direct mass measurements and global evaluation of atomic...

CER

N-T

HES

IS-2

018-

332

06/0

6/20

18N

NT

:201

8SA

CLS

151

Direct Mass Measurements and GlobalEvaluation of Atomic Masses

These de doctorat de l’Universite Paris-Saclaypreparee a l’Universite Paris-Sud

Ecole doctorale n◦576 particules hadrons energie et noyau : instrumentation, image,cosmos et simulation (PHENIICS)

Specialite de doctorat : Structure et reactions nucleaires

These presentee et soutenue a Orsay, le 6 juin 2018, par

M. WENJIA HUANG

Apres avis des rapporteurs :

M. YANLIN YEProfesseur des Universites, Peking UniversityM. STEPHANE GORIELYChercheur Qualifie FRS-FNRS, Universite Libre de Bruxelles

Composition du Jury :

M. DAVID LUNNEYDirecteur de Recherches, CSNSM (UMR8609) PresidentM. STEPHANE GORIELYChercheur Qualifie FRS-FNRS, Universite Libre de Bruxelles RapporteurMme HAO FANGPrincipal physicist, BIPM, France ExaminatriceM. YURI LITVINOVDirecteur de Recherches, GSI, Germany ExaminateurM. GEORGES AUDIDirecteur de Recherches, CSNSM (UMR8609) Directeur de these

Acknowledgements

Foremost, I would like to thank my supervisor Georges Audi for his continuous supportof my Ph.D. study. His patience, caution, and immense knowledge in experimental physicsinspired me the first day I came to the lab. Sitting next to him within 1-m distance allowedus to immerse in fruitful discussions for the whole day, from which I could learn almostevery detail in mass evaluation. Thanks for his encouragement and confidence in me, whichpushed me to try something new and took on challenges, which I had never been imaging Icould have done. I also appreciate his assistance in my daily life in France. I would also liketo express my gratitude to David Lunney, a man who stood at the forefront of physics andalways informed me of the latest news in mass measurements. He offered me the opportunityto work at ISOLTRAP/CERN, one of the greatest groups of mass measurements in theworld, a place in which I had ever dreamt of going when I came across mass measurementsfor the first time.

I want to thank all the members of the local group at ISOLTRAP, who provided mea warm welcome when I was there. Vladimir Manea, the backbone of ISOLTRAP, whodirected me and taught me everything he knew about Penning traps. I was always impressedby his knowledge and enthusiasm, and he was the man from whom I acquired the very firstimage of the ISOLTRAP setups. Frank Wienholtz, another backbone of ISOLTRAP andan expert at MR-TOF devices, who had always novel ideas and good suggestions. DinkoAtanasov and Maxime Mougeot, two “pythonic” guys, who provided help and suggestionsfor programming. I would also like to thank Andree Welker and Jonas Karthein for providinghelp and convenience when I was at CERN. I was so glad to work with these wonderful guys.Also many thanks to Vladimir and Frank, for their continuous support for data analysis andguidance after I finished my lovely stay at CERN.

In addition, I would like to thank all my CSNSM colleagues, who help me overcome thedifficulties I met in France. Staying with them also allowed me to know more about theFrench culture and broaden my horizon.

I also thank the people from Lanzhou. Yuhu Zhang, my master supervisor, for his basictraining in how to initiate research and support for my decision to work on Ame. MengWang, the coordinator of Ame, for his corrections and comments on my manuscript and hisfinancial support for me to attend international conference meeting in Beijing.

I would like to thank our collaborators in Ame : Georges Audi, Meng Wang, Filip Kondev,and Sarah Naimi, for their encouragement and insightful comments.

I would like to thank my jury members : Stephan Goriely from ULB, Yuri Litvinovfrom GIS, Hao Fang from BIPM, Georges and David from our local group, for their carefulreading and enlightening comments on my manuscript. Thanks Stephan Goriely and YanlinYe (Peking University) for their reports of the thesis. A special thanks goes to Stephan, forhis pointing out the defeat in my manuscript and hard questions.

I would like to thank China Scholarship Council for supporting my Ph.D. study ; and theENSAR program for its financial support for my stay at CERN.

Last but not least, I would like to express my sincere gratitude to my beloved wife, whoshowed confidence, faith in me and never complained if I had to work at weekends. Thanksmy parents for raising me up and their spiritual support, without which I could not havefinished my thesis. I am also grateful to my wife’s family for their support.

3

Preface

At the time I am writing this dissertation, I have spent more than three years and ahalf working on the Atomic Mass Evaluation (Ame). Ame is the most trustworthy, exclusiveresource related to the atomic masses. And the mass tables, the main product of Ame, arewidely used in the physics community. Aaldert Hendrik Wapstra, the founder and grandinquisitor of Ame, first noticed that the masses derived from different techniques can bebest deduced by a least-squares method. A. H. Wapstra, together with F. Everling, L. A.Konig, and J. H. E. Mattauch, provided such an evaluation process at the first InternationalConference on Nuclidic Masses (AMCO-1) in 1960. His philosophy still serves as a main dishfor the daily life of Ame. Georges Audi, the Guardian of Ame and also my Ph.D. supervisor,has been working in Ame since 1981. He is the reason why AME can continue in a healthyway until now. His passion for Ame also inspires me to take Ame as my Ph.D. subject.Nowadays, when we speak of the masses tables, one often refers to Audi-Wapstra’s masstables.

Ten mass tables have been published up to now, starting from the first version in 1961 tothe latest one in 2016. The only dissertation dedicated to Ame was accomplished by K. Bos inhis Ph.D. thesis in 1977 entitled “Determination of Atomic Masses from Experimental Data”,under the supervision of A. H. Wapstra. Since then, not only the experimental techniquesbut also Ame itself evolved.

The intention of this dissertation is not to cover every aspect in Ame (it is also impossibleto do so). The omissions are due to the limited length of the dissertation and my lack ofspecial knowledge in some domains. One could have followed a series of mass tables inwhich one could give emphasis at that time. The aim is instead to show to the readersthe most important features of Ame. I think it could be an opportunity for the public tograsp the basic concept of Ame and get acquainted with the way in which Ame treat thedata. In Chapter 1, a brief introduction to Ame and the related concepts will be given. InChapter 2, the indirect and the direct methods of mass measurements will be discussed,and two of the most important concepts of mass spectrometry, i.e., the resolving power andmass resolution, will be introduced. In Chapter 3, the philosophy of Ame will be illustratedin detail, together with a detailed example to show how Ame works. In Chapter 4, themost recent developments resulting from my Ph.D. work on the mass table, AME2016, willbe presented, such as the calculation of molecular binding energy, the energy correction ofthe implantation experiments, and the relativistic formula for the alpha-decay process. InChapter 5, the accuracy and the predictive power of different mass models will be discussed.In Chapter 6, the mass extrapolation of Ame will be introduced. In Chapter 7, the resultsfrom the Penning-trap mass spectrometry (ISOLTRAP) will be presented. In Chapter 8,the study of the systematic error at the ISOLTRAP multi-reflection time-of-flight massspectrometer will be discussed.

In order to finish this thesis, I consulted many Ame publications, from which I know howAme evolves. I think it is an essential process for each evaluator to know what Ame was inthe past so that we could have some hints of how to improve it for the future.

5

Contents

Acknowledgements 3

Preface 5

1 Introduction 15

2 Experimental Techniques 232.1 Energy Conversion . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Indirect Method . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Nuclear reaction . . . . . . . . . . . . . . . . . . . . . 242.2.2 Decay measurement . . . . . . . . . . . . . . . . . . . . 25

2.3 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Mass Resolution and Resolving Power . . . . . . . . . . 282.3.2 Mass Spectrometry . . . . . . . . . . . . . . . . . . . . 29

3 The Evaluation Procedures 353.1 General remark . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Introduction to least-squares method . . . . . . . . . . . . . . 363.3 Reduction of the problem . . . . . . . . . . . . . . . . . . . . 383.4 χ2 test and consistency factor . . . . . . . . . . . . . . . . . . 423.5 Flow-of-information matrix . . . . . . . . . . . . . . . . . . . . 443.6 Local adjustment in the lightest mass region . . . . . . . . . . 443.7 Removal of certain input data . . . . . . . . . . . . . . . . . . 48

4 Developments for AME2016 514.1 Molecular binding energy . . . . . . . . . . . . . . . . . . . . . 514.2 α- and proton-decay energies . . . . . . . . . . . . . . . . . . . 554.3 High precision α-decay energies . . . . . . . . . . . . . . . . . 614.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Mass Models 655.1 Semi-empirical approaches . . . . . . . . . . . . . . . . . . . . 655.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Predictive power . . . . . . . . . . . . . . . . . . . . . . . . . 71

7

CONTENTS

6 Mass Extrapolation 816.1 Regularity of the Mass Surface . . . . . . . . . . . . . . . . . . 816.2 Scrutinizing the Mass Surface . . . . . . . . . . . . . . . . . . 826.3 Subtracting a mass model from the experimental mass surface 83

7 Experiments 917.1 Principle of Ion Traps . . . . . . . . . . . . . . . . . . . . . . . 917.2 Experimental setup at ISOLTRAP . . . . . . . . . . . . . . . 957.3 Data analysis and discussions . . . . . . . . . . . . . . . . . . 96

8 MR-TOF MS 1078.1 Principle of MR-TOF MS . . . . . . . . . . . . . . . . . . . . 1078.2 Systematic error study . . . . . . . . . . . . . . . . . . . . . . 108

Conclusions and Outlook 125

Appendix A Relativistic formula of alpha decay 129

Appendix B TOF-ICR Spectra 133

Synthèse 155

8

List of Figures

1.1 Nuclear binding energy per nucleon for all known ground-statemasses from AME2016. . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Schematic representation of all the available nuclear data [Aud01]. 191.3 Number of publications included in Ame each year starting

from 1951 to the cut-off date in AME2016. The maximum num-ber appears in the year of 1995, where over 180 publicationsare included. It was due to the conference on Exotic Nuclei andAtomic Masses (ENAM-95), where mass measurements were acentral topic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4 Flow diagram of the four-phase computing process in Ame. . . 22

2.1 Valley of stability formed by black boxes (192 stable nuclides). 242.2 Diagram of all the common types of decay and reactions which

connect to a mass represented by a square, in which A is themass number, Elt the element symbol, N the neutron number,and Z is the atomic number. Letters from a to m in circlesrepresent different types of connection defined in Table 2.1. . . 27

2.3 Illustration of a) two barely separate spectral peaks of twomasses m1 and m2 with equal height and width, where two hor-izontal dash lines denote the height at 50% and 100%; b) twospectral peaks of the same masses as in a) but with unequalheight are mixed and cannot be separated. . . . . . . . . . . . 28

3.1 Connection plot with primary, secondary and unconnected items. 403.2 Connection plot in the 163Ho region. Each symbol (square and

circle) represents a nuclide: the large ones denote nuclides thatwould be used in the least-squares procedure; the small onesdenote secondary nuclides that would not be used in the least-squares procedure. The upper red square symbol indicates 163Dyand the lower one indicates 163Ho. . . . . . . . . . . . . . . . . 40

3.3 Birge Ratio of all the parallel data . . . . . . . . . . . . . . . 423.4 Connection plot of the lightest nuclides. Each line represents

an experimental datum. The corresponding reference papers arealso indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

9

LIST OF FIGURES

4.1 Illustration of α-decay spectra where line-1 and line-2 are cal-ibrants and line-3 is unknown at an equal distance from line-2. (a) Case for which the detector detects only the α-particleenergy. (b) Case where the detector detects also the recoilingnuclide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Detection efficiency K for different species at different recoilingenergy ER in Si-detector. The range of ER selected here coversmost of the decay experiment cases. . . . . . . . . . . . . . . . 58

5.1 Masses of different Sn isotopes calculated from different modelswith respect to the mass model DUZU. . . . . . . . . . . . . . 67

5.2 Root-mean-square deviations of eight mass models for nuclidesZ,N ≥ 8 with respect to mass tables AME2003, AME2012,and AME2016. The number in the parenthesis indicates thenumber of parameters in the corresponding mass model. Sameillustrations will be applied to the successive figures. . . . . . . 69

5.3 Root-mean-square deviations of eight mass models in four re-gions: Light (8 ≤ Z < 28, N ≥ 8), Medium-I (28 ≤ Z < 50)Medium-II (50 ≤ Z < 82) and Heavy (Z ≥ 82). . . . . . . . . 71

5.4 Root-mean-square deviations from AME2016 in four regions(represented by lines in green): (a) Light, (b) Medium-I, (c)Medium-II, and (d) Heavy. The Global rms deviation is alsodisplayed (represented by black lines). . . . . . . . . . . . . . . 72

5.5 δrms(2012) and δrms(new) in (a) Global (b) Light (c) Medium-I(d) Medium-II and (e) Heavy regions. . . . . . . . . . . . . . . 76

5.6 Display of deviations between masses from models and thatfrom AME2016 in color plots. . . . . . . . . . . . . . . . . . . 80

6.1 Screen shot of the “Interactive Graphical” tool displaying fourderivative of the mass surface: two-neutron separation energyS2n, two-proton separation energy S2p, α-decay energy Qα anddouble-beta-decay energy Qββ from upper left to bottom right.The lines between two points have the same iso-properties Z,N , Z and Z, respectively. A universal smoothness is identifiedin each quadrant, except when it comes across a shell closure:in the S2p plot at Z = 50 (a sudden drop of slope). . . . . . . 84

6.2 Two-neutron separation energies of Sn isotopes for all modelsdiscussed in Chapter 5. Two vertical dash lines signify magicnumbers N = 50 and N = 82. . . . . . . . . . . . . . . . . . . 85

6.3 Display of the differences between experimental masses and theDUZU model as a function of N and Z, of the experimentaltwo-neutron separation energies, and of the experimental two-proton separation energies in four quadrants. . . . . . . . . . . 86

10

LIST OF FIGURES

6.4 Four derivatives the same as Fig. 6.3 but zooming around 79Cu.New extrapolation (in blue) based on the results from [WAA+17]. 88

7.1 Hyperbolic electrode geometry of Paul trap (a) and Penningtrap (b). Trapping of charged ions can be realized by applyinga voltage difference between the ring electrode and the end elec-trodes. Penning traps can also have cylindrical electrodes (c).Figure from [Bla06]. . . . . . . . . . . . . . . . . . . . . . . . 92

7.2 Sketch of ion motion in a Penning trap. . . . . . . . . . . . . . 937.3 Illustration of the TOF-ICR technique . . . . . . . . . . . . . 957.4 Schematic view of the ISOLTRAP setup [KAB+13]. See text

for details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.5 Time-of-flight spectrum of 57Cr. . . . . . . . . . . . . . . . . . 1007.6 TOF-ICR spectra for 168Lu. Two excitation times are taken:

Trf = 1.2 s (a) and Trf = 3 s (b). The arrows indicate theposition of the expected ground state. . . . . . . . . . . . . . . 102

7.7 Two-neutron separation energy in the ytterbium region between(Ho (Z = 67) and W (Z = 74)). The experimental data are de-noted by black circles, estimated masses are denoted by emptydiamonds, and the red circle represents the new 178Yb mass. . 103

7.8 Flow of information diagram for the chromium masses fromA = 52 to A = 55. Each box represents a nuclide, with the massuncertainty (in keV) in the lower right corner. The numbers inblack represent the old evaluation in AME2012, and numbers inblue represent the new evaluation including the new chromiumresults. The numbers in blue in the lower parts indicate theinfluences of the current data on the corresponding nuclides.The dash arrows indicate the contribution from other experiments.106

8.1 Time of flight of five reference species fitted as a function of thenumber of reflection. . . . . . . . . . . . . . . . . . . . . . . . 109

8.2 Illustration of the relative residuals as a function of reflectionnumber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.3 Relation between the square root of mass and the TOF atN = 0(a). Residuals of TOF at N = 0 (b). . . . . . . . . . . . . . . . 111

8.4 Mass of 87Rb determined by using 39K and 133Cs as referencemasses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.5 Standard deviation of 100 TOF spectra for each species at dif-ferent number of reflection. . . . . . . . . . . . . . . . . . . . . 114

8.6 Relative residuals for three reference ions. The data was aver-aged by 100 spectra. . . . . . . . . . . . . . . . . . . . . . . . 114

8.7 Mass determination of 87Rb from 100 spectra using 85Rb and133Cs as reference ions. . . . . . . . . . . . . . . . . . . . . . . 115

11

LIST OF FIGURES

8.8 Mass of 87Rb determined by varying the number of the accu-mulated spectra at different reflection number. . . . . . . . . . 115

8.9 Scan of Ulift voltages of 85Rb at N = 900 from 1000 V to1080 V in steps of 4 V (lower figure). Beam intensity as afunction of Ulift. . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.10 Optimized cavity voltage as a function of reflection number Nof 85Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.11 Mass determination of 87Rb at the optimum cavity voltage fortwo different data set. . . . . . . . . . . . . . . . . . . . . . . 118

8.12 (a) Mass of 87Rb determined at reflection number 1000. The x-axis indicates when the measurement was performed. (b) Distri-bution of deviations of (a). The red curve represents a Gaussianfit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.13 Time-of-flight spectrum of A = 46 nuclides. . . . . . . . . . . . 1218.14 Mass deviations from AME2016 for 47 measurements. The in-

sert figure displays the distribution of the deviation divided bythe uncertainty for each measurement. . . . . . . . . . . . . . 122

8.15 Parameter β in Eq. 8.2 for on-line measurements. . . . . . . . 1238.16 Difference of mass of the same nuclide determined by using one

and two reference ions. The red curve is the linear fit to thedata point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

B.-2 TOF-ICR spectra. . . . . . . . . . . . . . . . . . . . . . . . . 136

12

List of Tables

2.1 Calculations of common types of decay and reaction energies. . 26

3.1 Typical input in Eq. 3.2 and corresponding mass equation. . . 373.2 Related information for the local adjustment in the lightest

mass region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Adjusted Q-values in the lightest mass region. v/s indicate the

difference between the input Q-value and the adjusted Q-valuedivided by the uncertainty of the input Q-value. . . . . . . . . 48

4.1 Frequency ratios and mass differences. . . . . . . . . . . . . . 544.2 Accurate α-decay calculations. Col. 1 is the decay incident ,

Col. 2 is the α energy, Col. 3 and Col. 4 are the calculated decayenergies using classical (Eq. 4.12) and relativistic (Eq. 4.21)formulae, respectively, and col. 5 is difference between the valuesfrom two formulae. . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1 Information of eight mass models: the year of publication, thenumber of parameters in the model, and the mass table thatwas used to fit parameters. . . . . . . . . . . . . . . . . . . . . 66

5.2 Root-mean-square deviation δrms, mean deviation δ, and max-imum deviation δmax for nuclides with Z,N ≥ 8 with respectto three mass tables AME2003, AME2012, and AME2016. Theresults for the eight mass models are listed. . . . . . . . . . . . 68

5.3 Root-mean-square deviation δrms, mean deviation δ, and maxi-mum deviation δrms from AME2016 in four regions: Light (8 ≤Z < 28, N ≥ 8), Medium-I (28 ≤ Z < 50), Medium-II (50 ≤Z < 82), and Heavy (Z ≥ 82). . . . . . . . . . . . . . . . . . . 70

5.4 Root-mean-square deviations in the Global (Z,N ≥ 8), Light(8 ≤ Z < 28, N ≥ 8), Medium-I (28 ≤ Z < 50) Medium-II(50 ≤ Z < 82) and Heavy (Z ≥ 82) regions. The rms deviationsare calculated separately for the masses that were known inAME2012 δrms(2012) and for the new ones in AME2016 δrms(new). 73

13

LIST OF TABLES

7.1 Experimental details in the production of the ions of interest.Listed are the experiment date, the target, the ionization tech-nique, the ion energy from ISOLDE, and the mass separatorused. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.2 Frequency ratios between the ions of interest and reference ions. 997.3 Influences of the ISOLTRAP results and the adjusted chromium

masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.1 Fitting parameters a and b in Eq. 8.7 . . . . . . . . . . . . . . 110

14

Chapter 1

Introduction

When we speak of mass, one would immediately think of the mass of theSun, our body, a coin, and the like. However, mass, no matter how small it is,is a quantity that every existing object possesses. We can: weigh our body byscales; weigh the mass of a coin by counterpoise balance; we can even weigh themass of the Sun by Newton’s law of universal gravitation through observingthe period of the Earth moving around the Sun. But how to weigh the massof an atom, whose mass and size are extremely smaller than those of the Sunsince no one can really see it?

The history of atomic-mass measurements is as old as nuclear physics (onecan refer to [Aud06] for the history of early atomic-mass measurements). In1897, J. J. Thomson found that the cathode rays containing electrical chargeshad a very large value for the charge-to-mass ratio. He measured this ratio byusing electric and magnetic fields and tracking their trajectories. This historicevent marks the discovery of what we call “electron” today. In 1912, withthe development of Thomson’s instrument, F. W. Aston first showed evidencefor the presence of two different isotopes of neon, having mass numbers of20 and 22, respectively. During his career, Aston discovered more than 200naturally-occurring isotopes, which were the first systematic studies of atomicmasses. Aston found that the hypothesis: “the mass of all isotopes were integermultiples of that of hydrogen” was very nearly true (He replaced the originalword “atom” by isotope). This hypothesis was what we call the whole-numberhypothesis, put forward by J. Dalton andW. Prout. But very nearly true meanssomething is missing. Actually, Aston found that the masses he measuredfor all stable isotopes were less massive than that from the whole-numberhypothesis. This famous “mass defect” is henceforth explained as the “bindingenergy” effect, based on Einstein’s famous mass-energy equation E = mc2.

Nowadays, we known that the basic building blocks of a nucleus are pro-tons and neutrons ∗ (they are two types of nucleon). The “mass excess”, atthe origin called “mass defect”, is defined as the difference between the atomic

∗. Neutron was discovered by J. Chadwick in 1932.

15

mass of a nucleus and its mass number:

ME = M − A · u, (1.1)

where u stands for the Unified Atomic Mass Unit (defined below). The “massexcess” is a more convenient way to tabulate atomic masses, since it carriesless digits.

Nuclear Binding EnergyThe mass of an atomM(N,Z) is the sum of the masses of its constituents

(protons, neutrons, and electrons) minus its electronic binding energy andnuclear binding energy:

M(N,Z) = Nmn + Zmp + Zmec2 −

Z∑i=1

Bi −Bnuc(N,Z), (1.2)

where mn is the neutron mass, mp the proton mass, me the electron mass,Bi the i-th electron binding energy, and Bnuc the nuclear binding energy. Thetotal electronic binding energy of a hydrogen atom is 13.6 eV and can reach10 ∼ 100 keV for heavy nuclides. Considering that the atomic mass is of theorder of A × 1000 MeV, a precision better than 1 part in 1010 is required tostudy the atomic effects [BNW10]. For the nuclear binding energy Bnuc(N,Z),or more interestingly the nuclear binding energy per nucleon which is of theorder of 8 MeV, a precision of 1 part in 106 is needed to study nuclear shelleffects [LPT03]. In this chapter, we concentrate only on the nuclear bindingenergy and without making a confusion, Bnuc(N,Z) is replaced by B(N,Z).

The nuclear binding energy of a nucleus is traditionally expressed as:

B(N,Z) = Nmn + ZM(1H)−M(N,Z), (1.3)

where M(1H) is the mass of hydrogen. The mass, or equivalently, the nuclearbinding energy, includes all the interactions (strong, weak, and electromag-netic) at work in the nucleus. The systematic studies of the binding energyprovide valuable clues for nuclear structure.

Fig. 1.1 shows the nuclear binding energy per nucleon for 2498 knownmasses in their ground states [WAK+17]. Several phenomena can be seen in thefirst place. The curve is relatively constant at B/A ∼ 8 MeV, except for lightnuclides. This leads to the idea of saturation of nuclear forces, i.e., each nucleoninteracts only with its neighboring nucleons (otherwise B would increase asa function of A2 instead of A). Secondly, the curve reaches its maximum atA ≈ 60 (62Ni is the mostly bound nucleus with B/A = 8.794 MeV and isfollowed by the second mostly bound nucleus 56Fe which has 8.790 MeV),meaning that there are two ways to gain energy: either below A = 60 by

16

CHAPTER 1. INTRODUCTION

Figure 1.1: Nuclear binding energy per nucleon for all known ground-statemasses from AME2016.

assembling light nuclides into a heavier one, or above A = 60 by breaking aheavy nuclide into light parts. The first one is called nuclear fusion and thesecond one is called nuclear fission.

In trying to describe the curve in Fig. 1.1 a semiempirical mass formulacan be derived:

B = avA− asA2/3 − acZ(Z − 1)A−1/3 − asym(N − Z)2

A+ δ, (1.4)

where av, as, ac, asym, and δ are five coefficients to be determined. This formulawas first devised by Weizsäcker [Wei35] and Bethe [BB36], which serves alsoas a basic ingredient in some modern mass models. The first term is calledthe volume term and comes from the fact that the binding energy per nucleonis linear to the mass number A. The second term is called the surface termand stems from the fact that the nucleons on the surface interact only withthe internal nucleons. The third term is the Coulomb term which accounts forthe Coulomb repulsion between protons. The fourth term is based on the factthat nuclei with N ≈ Z are more stable than their neighbors. The last termcomes from the pairing force, which has the tendency to couple nucleons ofthe same type to stable configurations (zero total angular momentum). Thispairing energy δ can be expressed [KH88] as +apA

−3/4 for even Z and N ,−apA−3/4 for odd Z and N , and zero for odd A nuclides.

The five terms of the Bethe-Weizsäcker formula in Eq. 1.4 describes a nu-cleus “macroscopically”, which mimic a charged droplet. The Bethe-Weizsäcker

17

mass formula is the mostly used nuclear mass model, which lies in its simplic-ity and relatively good predictive power for most of the nuclei. However, itfails to describe the nuclei with “magic” numbers (N and Z equal 2, 8, 20, 28,50, and 82), meaning that extra components (such as shell effect) should bealso considered in the formula.

Unified Atomic Mass Unit

Nowadays, the standard mass unit u is defined as one-twelfth of themass of a carbon-12 atom in its nuclear and atomic ground state, namely1u † = M(12C)

12. The unit “u” stands for “Unified Atomic Mass Unit”, which has

its historical reason. There existed, before 1960, two scales on atomic masses:taking the mass of one atom 16O as 16 units (physics scale) or taking themass of natural-mixture oxygen sample as 16 units (chemical scale or atomicweight). The proposals for unifying the mass unit were widely discussed, e.g.,J. H. E. Mattauch was one of the physicists who had been studying the scaleproblem at that time. In 1956, A. Nier and A. Ölander suggested that atomicweight scale be based on a 12C atom to the International Commission onAtomic Weights. It had been concluded that 12C was not only an acceptablesubstitute but also had operational advantages for physical comparisons ofnuclidic masses. However, the proposal for unifying the mass unit did notplease chemists. As demonstrated by T. P. Kohman, J. H. E. Mattauch, andA. H. Wapstra [KMW58], accepting the new definition of the mass standardwould cause a change of 275 ppm instead of using the oxygen atomic weight. Itmeant chemists would lose millions of dollars selling their products. A revisionof innumerable tabulations of data would also need to be initiated. Despiteall the difficulties in unification, the candidate 12C is advantageous for severalreasons. First, the mass of a nuclide can be expressed very nearly to its massnumber A (maximum deviation of 0.1 u). Secondly, 12C was the most impor-tant substance in the mass-spectroscopic determination of nuclidic masses andit will allow direct comparisons with standard mass. Moreover, carbon formsmany hydrides at almost every mass number (up to A = 120). The atomicmass unit based on 12C was approved by the International Union of Pure andApplied Physics (1960) and is still used worldwide.

Nuclear Data

Nuclear data, such as atomic mass, excitation energy, half-life, magneticmoment, spin and parity, etc., finds its application from basic research suchas nuclear structure studies, astrophysics, and fundamental physics studies, tonuclear engineering, medicine, environment, and the like. All these domains

†. 1 u = 1.660539040(20)× 10−27 kg

18


Figure 1.2: Schematic representation of all the available nuclear data [Aud01].

require a canonical data bank which stores all the information and providesrecommended values for users.

We can imagine that all the nuclear data are stored in a building, asillustrated in Fig. 1.2. Each vertical bar represents a nuclide, whose ground-state properties are stored on the ground floor and the properties of excitedstated are stored above the ground floor.

The Evaluated Nuclear Structure Data File (ENSDF) contains the evalu-ated nuclear properties of all known nuclides derived from nuclear reaction anddecay measurements [Tul96]. The ENDSF files are organised by mass numberA and this A-chain evaluations are undertaken by the members of the Interna-tional Nuclear Structure and Decay Data (NSDD) network. Such evaluationis vertical, since all nuclides within the same mass chain are evaluated at oncewhen new experimental information becomes available [NBD+17].

Ame is a bit different from ENSDF, in that it connects a quantity whichmaps across the whole nuclear chart: the mass. Thus the evaluation of atomicmass is horizontal. All the indirect and direct methods (discussed in Chapter 2)yield the mass difference between two or more nuclides, which result in anoverdetermined system for mass.

The Ame process

Ame shares many similarities with other nuclear-data evaluation projects[Aud01]. The first step in the evaluation is to make a compilation, i.e., collectall available data. Up to the publication of AME2016 [WAK+17], we scanned24 different kinds of physics journals and conference proceedings. Each year,around 100 publications are included in Ame. Fig. 1.3 shows the number of

19

publications included in Ame yearly until the cut-off date of AME2016.

Figure 1.3: Number of publications included in Ame each year starting from1951 to the cut-off date in AME2016. The maximum number appears in theyear of 1995, where over 180 publications are included. It was due to the con-ference on Exotic Nuclei and Atomic Masses (ENAM-95), where mass mea-surements were a central topic.

The second step is the critical reading process. Decay spectroscopy, re-actions, and mass spectrometry determine masses using different techniquesand give very diverse uncertainties. During critical reading, we should make ajudgment on several parts [Aud01]:• calibration procedure: the use of calibrants, calibration function;• spectra examination: peak position, shape, intensity, the goodness of

fit;• the primary data: keep only the original data and not use other val-

ues that are deduced by the combination with other experiments andevaluated values.

After examining a paper, we will compare the new results with the previ-ous ones, if any, and their quality will be judged (see Section 3.7 in Chapter 3).

The last step is to enter the new data in the Ame database. Here is anentry associated with the α-decay energy of 236Cm in Ame:

236 850963000a1 UHGSa 10Kh06 7074.1 20. 236Cm(α)232Pu 6954 20 A,

20


where the first 14 digits is the unique ID-number for each input datum, fol-lowed by a label U (means no weight for this entry), a reference for the labora-tory “GSa", and its Nuclear Science References (NSR) [PBK+11] key-number“10Kh06”, the α-decay energy with its associated uncertainty in keV, the inputequation, the α-particle energy with its uncertainty in keV, and a label “A"which denotes that the decaying level and the final level are well established.If they were not, we should have given the label “O", causing the program toincrease the uncertainty to 50 keV.

The Ame computer program

The Ame evaluators work primarily on three files: Q-file, M -file, and theR-file. The Q-file contains a wealth of data which has the same format asthe example discussed above. The M -file contains the fundamental propertiesof nucleus, such as atomic mass, excitation energy, half-life, spin and parity,decay mode, etc. The R-file includes all the references related to the inputdata.

After data compilation, the Ame computer program will perform a four-phase task [Aud01]:

1. Decoding and Checking. The data will be decoded and the correctnessof the ID assignment for each datum will be checked.

2. Building Connection. The connection between the masses will be built,thus allowing the separation between primary and secondary (see defi-nitions in Section 3.3) nuclides.

3. Applying the least-squares method. See Chapter 3 for details.4. Calculating outputs. Different sorts of outputs will be represented, in-

cluding the adjusted masses, adjusted input values, flow-of-informationmatrix, etc (see Chapter 3 for details).

Once finishing the four-phase program, one can compare the input datawith the adjusted one, from which we can recognize if there exists inconsistencyor not, or the labels for some data should be reassigned. For example, thevalues of some input data can be obtained by the combination of other datawith higher precision. In this case, the less precise data will be labeled “U".This is the very essential part of evaluation. After such fine tuning, whichwould not affect the adjusted results, the figures for separation energy anddecay energy can be provided. The computing process is displayed in a flowdiagram in Fig. 1.4.

21

Figure 1.4: Flow diagram of the four-phase computing process in Ame.

22

Chapter 2

Experimental Techniques

2.1 Energy Conversion

Mass measurements can be carried out by two different methods: theindirect method, which establishes an energy relation between two or morenuclides through reactions or decays, and the direct method, which “observes”the inertial mass from its movement in an electromagnetic field. The energyrelation established in the indirect method is expressed in electron-volt (eV),while the masses determined by the direct method is expressed in “unifiedatomic mass” unit (u).

The choice of the volt in the energy unit is not evident: it can be expressedin volt (VSI), which is based on the internationally accepted definition (SI),or the one as maintained (V90) by the Bureau International de Poids et Mea-sures (BIPM) using the Josephson effect [Jos62]. It was demonstrated [CW83]that the energy would be expressed more precisely in the maintained voltthan in the standard volt. The relation between the two defined volts andtheir relations with the atomic mass unit can be expressed as [MNT16a]:

V90 = 1 + 9.83(61)× 10−8 VSI

1 u = 931494.0954± 0.0057 keVSI

1 u = 931494.0038± 0.0004 keV90.

When combining the inertial mass from mass spectrometry and the energyrelation from reactions and decays, one has to apply the conversion factor.The last equation, due to its higher precision, was used in AME2016 for allthe energy conversion. For simplicity, the label “V” denotes the maintainedvolt in the following text.

23

2.2. INDIRECT METHOD

Figure 2.1: Valley of stability formed by black boxes (192 stable nuclides).

2.2 Indirect Method

2.2.1 Nuclear reaction

From Einstein’s Mass-Energy equation E = mc2, we known that thereleased energy in a reaction is directly related to the involved masses. For anuclear reaction type A(a, b)B, the released (absorbed) energy is defined as:

Q = MA +Ma −Mb −MB. (2.1)

The reaction is exothermic if Q > 0 and endothermic if Q < 0. In most casesthe masses of the target A, the projectile a, and the ejectile b are well known.Hence the mass of the fragment B can be derived by the measured Q-value,based on reaction kinematics. Generally, the target, the projectile, and theejectile are stable or very close to stability, and the fragment will not be veryexotic. In the nuclear chart, 192 nuclides are stable, meaning that no decaymode is observed in experiments. These nuclides form the valley of stability,see Fig. 2.1, which is the destiny of all the unstable nuclides.

In the 1970s, the advent of radioactive ion beams allowed to explore theproperties of nuclides with extreme proton-to-neutron ratio [BND13]. The bestway to study the properties of these exotic nuclides in reaction is the missingmass method [Pen01]. For a binary-product system, the property, here themass, of a nuclide can be determined by measuring the spectrum of its well-known partner from the laws of energy and momentum conservation. Thismethod can be applied even if the nuclide under investigation has no bound

24

CHAPTER 2. EXPERIMENTAL TECHNIQUES

state. For example, the mass of 7He (with half-life 2.51 zs ∗) was determinedby the 9Be(15N, 17F) reaction [BBG+99]. However, the missing-mass methodwould meet difficulty when it goes further from stability due to the necessityof finding a stable complementary nuclide.

An alternative to the missing mass method is to measure unbound nu-clides using invariant-mass spectrometry. For example, if we want to know themass of an unbound nuclide B = x + C (where x and C are two decayingproducts), we can measure the four-momentum of the subsystem x + C. Thedecay energy, which is a quantity that is measured directly, of the unboundnuclide B is related to other masses through:

Q = M(B)−M(x)−M(C), (2.2)

where M(x) and M(C) are the masses of the decaying particles. The advan-tage of this method is that the energy of the incident beam does not needto be known in advance. However, this method may suffer from low efficiencyfor the registration of light particles, such as neutron, which is the most com-mon decaying particle for neutron-rich nuclides, and the difficulty in assigningground-state. Some of the light exotic nuclides approaching the neutron dripline were measured this way, such as 10He [JAA+10, KSB+12], 13Li [KLD+13],16Be [SKB+12], and 26O [KNT+16].

Neutron-induced γ-ray measurements AX(n,γ)A+1X give directly informa-tion on the one-neutron separation energy through:

Sn = M(AX) +Mn −M(A+1X). (2.3)

Since the γ-ray energies are usually measured with high precision, the (n, γ)reactions, along with proton capture reactions, serve as backbone † in Ame.For example, the γ-ray energy for the reaction 185Re(n, γ)186Re was measuredwith a precision of 0.05 keV (σ(m)/m = 3×10−10) [MLH+16] at the BudapestResearch Reactor [Ros02].

2.2.2 Decay measurement

In nuclear-decay experiments, the decay energy is often measured to de-termine an unknown mass if the mass of its decay-companion is known. Betadecay is the universal phenomenon for all particle-bound nuclides far from sta-bility. Since the beta-decay spectrum has continuous distribution, the decayenergy is obtained from the maximum energy so-called the endpoint energy.For neutron-rich nuclei, beta decay occurs by emitting an electron (β−), wherethe β−-decay Q-value is associated with the mass difference Qβ− = MA−MB.

∗. 1 zs = 1× 10−21 s†. Another contribution to the backbone comes from Penning-trap mass spectrometry.

25

2.2. INDIRECT METHOD

Table 2.1: Calculations of common types of decay and reaction energies.

Q(β−) = M(A,Z)−M(A,Z + 1) (a)Q(2β−)= M(A,Z)−M(A,Z + 2) (b)Q(4β−)= M(A,Z)−M(A,Z + 4) (c)Q(β−n)= M(A,Z)−M(A− 1, Z + 1)−n (d)S(n) =−M(A,Z) +M(A− 1, Z)+n (e)S(p) =−M(A,Z) +M(A− 1, Z − 1)+1H (f)Q(εp) = M(A,Z)−M(A− 1, Z − 2)−1H (g)S(2n) =−M(A,Z) +M(A− 2, Z)+2n (h)Q(d,α) = M(A,Z)−M(A− 2, Z − 1)−2H−4He (i)S(2p) =−M(A,Z) +M(A− 2, Z − 2) + 21H (j)Q(p,α) = M(A,Z)−M(A− 3, Z − 1)−4He+p (k)Q(n,α) = M(A,Z)−M(A− 3, Z − 2)−4He+n (l)Q(α) = M(A,Z)−M(A− 4, Z − 2)−4He (m)

Proton-rich nuclei decay via the emission of a positron (β+), where the decayenergy Qβ+ = MA−MB − 2mec

2, or via electron capture (EC) process, whereQEC = MA −MB.

Beta-decay spectrometry has long been used as a powerful tool to studythe properties of nuclei not too far from stability. When further out of stability,due to high energy available, such experiments could suffer from the Pande-monium effect [HCJH77], where the decaying daughter nuclide is populatedthrough a large number of excited levels from which the energy of the emittedγ is not observed.

For the β+ decay, it would also suffer from the summing effect where apositron annihilates into two photons. For the EC process, the Q-value is nota quantity that can be measured directly. One has to rely on the measurementof the X-ray intensities, the correct assignment of the decay-level schemes andtheoretical calculations.

The measurement of the α-decay energy Qα yields the mass of a heavynuclide through: Qα = M(N,Z) − M(N − 2, Z − 2) − M(4He). If the α-decay chain ends up with a nuclide with known mass then we can deduceall the masses along the chain from the succession of α lines. For even-evennuclei, ground state to ground state transitions dominate and the assignmentis usually reliable. For odd-odd nuclei, on the contrary, their α decay does notgo directly to the ground state for most of the cases.

The most accurate α-decay energy came from magnetic spectrograph ofthe Bureau International des Poids et Mesures [GR71] and such high-precisionα energies serve as standards for other α-decay experiments. Nowadays, theimplantation method has been applied widely to measure α-particle energiesof heavy nuclides using semiconductor (e.g. silicon) detector. In such case, the

26


partial deposited energy from the heavy recoil should be considered (see Sec-tion 4.2 for detailed discussions).

The masses of nuclides close to the proton-drip line between Z = 50 andZ = 80 have been obtained mainly from proton radioactivity [BB08]. Thisspecial decay mode allows to investigate the properties of proton-rich nucleiwhich is of interest to nuclear and astrophysics models [TBH+12].

Table 2.1 lists, for a specific nuclide (A,Z), the derived values for differenttypes of decay, separation, and reaction energy as the combinations of atomicmasses (the letters correspond to the connection diagram in Fig. 2.2).

Figure 2.2: Diagram of all the common types of decay and reactions whichconnect to a mass represented by a square, in which A is the mass number,Elt the element symbol, N the neutron number, and Z is the atomic number.Letters from a to m in circles represent different types of connection definedin Table 2.1.

2.3 Direct MethodsInstead of measuring the released energy in reactions or decays, which

could involve the identification of complex level schemes, the mass can be

27

2.3. DIRECT METHODS

measured directly by mass spectrometry. Mass spectrometry is an analyticalmethod that determines the mass-to-charge ratio of ions. Before discussingdifferent types of mass spectrometers, two important concepts, i.e., mass res-olution and resolving power, should be introduced. They will allow a betterunderstanding of the performance of the different setups.

2.3.1 Mass Resolution and Resolving Power

The relation between mass resolution and resolving power is symbiotic:resolution is the experimental observable of the resolving power of an instru-ment and resolving power is the ability of an instrument to separate two spec-tral peaks which are similar in mass-to-charge ratio.

The mass resolution ‡ is conventionally defined by the Bureau Interna-tional des Poids et Mesures (BIPM) as the minimum distinguishable distancebetween two peaks of equal height and width. Marshall et al. [MBC+13] definesthe resolution in the same way that there exists a detectable “valley” betweenthe two close peaks. Fig. 2.3 (a) shows two peaks that are barely separated bya detectable valley. If the distance between the two peaks is smaller than thefull width at half-maximum (FWHM) of the peak height ∆m50%, they wouldnot be separated. Generally, unit resolution can separate mass 50 from mass51, or 100 from 101; resolution of 0.01 u is needed to distinguish mass number100 from 100.01, etc. In order to separate two spectral peaks with differentintensity, as illustrated in Fig. 2.3 (b), the resolution should be improved. Asdemonstrated in [MBC+13], the required mass resolution should be ∼ 10 timessmaller for two peaks with equal width but with height ratio of 100:1.

Figure 2.3: Illustration of a) two barely separate spectral peaks of two massesm1 and m2 with equal height and width, where two horizontal dash linesdenote the height at 50% and 100%; b) two spectral peaks of the same massesas in a) but with unequal height are mixed and cannot be separated.

‡. If presented without unit, it means relative resolution.

28


Conventionally, the mass resolving power is defined as:

R =m1

m1 −m2

,

where m1 and m2 denote the heavier and lighter mass, respectively. In a spec-trum where exists only one single peak, such definition is still applicable. Inthis case, the mass resolving power is m/∆m50%. Higher resolving power in-dicates a better ability to distinguish two peaks with smaller mass difference.In the example given above, to separate two peaks at mass 100 and 100.01 ofequal height, a spectrometer with a resolving power of 10,000 is required atleast.

One would notice that the BIPM definitions of mass resolution and massresolving power are in contradiction to the definition given by the Interna-tional Union of Pure and Applied Chemistry (IUPAC) [IUP14], where bothare defined as mass ratios in mass spectrometry. In this thesis, the definitionby BIPM is adopted.

2.3.2 Mass Spectrometry

All direct methods (except the MR-TOF mass spectrometry) describedbelow measure the motion of a charged ion in a magnetic field [LS01]:

Bρ

v= γ

m

q, or γ

m

q=B

ωc, (2.4)

where Bρ is the magnetic rigidity of the charged ion, v the velocity, γ theLorentz factor, m the rest mass, q the charge state, B is the strength of themagnetic field, and ωc is the cyclotron angular frequency. In principle, onecan measure two or three parameters in absolute values to obtain the mass.However, such absolute measurements are limited by the precision of the de-vices and are usually not practical. To avoid the direct measurements of theabsolute values of the apparatus, a delicate calibration is imperative.

TOF-Bρ mass spectrometry

The time-of-flight-magnetic-rigidity (TOF-Bρ) technique offers a good op-portunity to map a wide range of exotic and short-lived nuclides. The atomicmass of an ion can be deduced from Eq. 2.4 by measuring the time of flight(TOF) in a magnetic field [MG13]:

Bρ =γm

q(

L

TOF), (2.5)

where L is the flight length. Usually, the time of flight can be measured pre-cisely while the magnetic rigidity and flight length are unknown. The mass of

29

2.3. DIRECT METHODS

an ion of interest can be determined by using well-known masses as calibrants,from which the relation between the time of flight, magnetic rigidity and themass can be deduced.

The TOF-Bρ technique has been used mainly at three facilities: the en-ergy loss spectrometer (SPEG) [BFG+89] at the National Heavy Ion Labora-tory (Ganil), the time-of-flight isochronous spectrometer (TOFI) [WVW+85]at Los Alamos National Laboratory (Lanl), and the most recent one at theNational Superconducting Cyclotron Laboratory (Nscl) [MES+12].

The SPEG spectrometer measureed the time of flights in a typical range1000-1500 ns over a 116-m flight path [BFG+89]. The time of flight is measuredusing a radio-frequency signal of the accelerator as the start signal and the stopsignal was provided by a plastic detector with resolution of 350 ps (FWHM). Toaccount for the velocity dispersion, which introduced the deviation in the flightpath, the position was measured by two drift chambers for vertical position,and a position-sensitive parallel plates counter, for horizontal position. Theposition measurement provided a resolution of 10−4 in the determination ofBρ. Combining with the resolution of 2.9× 10−4 in the TOF measurement ata given time of flight 1200 ns, the final mass resolution was 3.2 × 10−4. Theexperimental masses were obtained using a fit function up to second order inmass number and atomic number [GMO+12].

TOFI employed an isochronous design [WVW+85], i.e., an ion with agiven mass-to-charge ratio travels along a longer path with higher velocitybut a shorter path with lower velocity, which resulted in a time of flight de-pending only on the mass-to-charge ratio. TOFI measureed a typical timeof flight of 600 ns with a flight path of about 14 m. A time resolution of230 ps (FWHM) [VWV+86] was obtained by using microchannel plate (MCP)detectors, which resulted in a mass resolution of 3.8× 10−4. The mass of ionsof interest were obtained by fitting a quadratic function to the measured timeof flight of the known masses.

The NSCL spectrometer shares many similarities with the SPEG spec-trometer. The flight path is about 60.6 m with a typical time of flight of about500 ns [MGA+15]. The time of flight is measured by two plastic scintilla-tors and the magnetic rigidity is measured with position-sensitive microchan-nel plate detectors. The time resolution is 80 ps ,which includes the positionresolution of the MCP detector, based on which a mass resolution of about1.6× 10−4 [MES+12] is obtained.

The TOF-Bρ technique can access the most exotic nuclides with half-lifedown to 1 µs due to its short flight path and hence short measuring time. Thesimultaneous production of the less exotic nuclides at the same experimentscan also provide reliable masses for calibration. These two features enable theTOF-Bρ technique to be suitable for short-lived, exotic nuclides.

However, this technique cannot in most cases separate a ground state froman eventual (long-lived) isomeric state of a nuclide due to its limited resolvingpower. For example, a resolving power of 1 × 104 is needed to separate the

30


ground state and its isomeric state at excitation energy of 3 MeV for a nuclideat A = 30, which is beyond the capability of all the TOF-Bρ spectrometersdiscussed above. Moreover, neither the magnetic rigidity nor the flight lengthcan be measured with sufficient precision, which means that an unknown massis usually deduced from a complex calibration function using as many referencenuclides as possible. This could lead to erroneous results if some of the low-lying excited states have not been identified.

Storage Ring Mass Spectrometry

The idea of increasing of the flight path and thus the resolving power opensa new era for TOF mass measurements. The circulating ions are confined in awell-defined orbit by a magnetic field and their revolution times or frequenciescan be determined with various methods. There exists only two facilities whichperform mass measurements using storage rings: the experimental storage ringESR [Fra87] at GSI and the experimental cooler storage ring CSRe [XZW+02]at the Institute of Modern Physics (IMP), Lanzhou. Both facilities can storeand measure ions at relativistic energy.

The difference in revolution frequency of two ions is related to the differ-ence in mass-over-charge ratio through [FGM08]:

∆f

f= − 1

γ2t

∆(m/q)

m/q+ (1− γ

γ2t

)∆v

v, (2.6)

where γ is the Lorentz factor, γ2t the transition point energy of the storage

ring, and v is the velocity of the ion.In order to minimize the dependence on the velocity term in Eq. 2.6

and obtain a direct expression of mass in terms of revolution frequency, twocomplementary methods can be used [FGM08]. One is called the SchottkyMass Spectrometry (SMS) and the other is the Isochronous Mass Spectrometry(IMS). In the SMS mode, the velocity dispersion is reduced to a level of wellbelow ∆v

v∼ 10−6 by using electron cooling [FGM08]. The cooling process lasts

for several seconds and hence SMS is only applicable for long-lived nuclides.The beam noise is picked up by two opposite metallic plates and the timesignals are Fourier tranformed to frequencies. Most of the frequency spectraare measured around the 30th harmonic [FGM08] in order to have a bettersignal-to-noise ratio.

In the IMS mode, no electron cooling is needed, and instead, by tuningthe optical setting γt = γ, ions with different velocities will have differentorbit lengths resulting in the same revolution time. Without cooling, IMS canmeasure ions with half-lives down to tens of µs, which is equivalent to tensof revolution. In IMS, a thin carbon foil is placed inside the storage ring andthe secondary electrons induced by the passing ions are recorded by a MCPdetector.

31

2.3. DIRECT METHODS

Both methods are employed in ESR and the resolving power of 7.5× 105

was achieved by SMS [GAB+01] and 2 × 105 by IMS [SKL+08]. CSRe usespresently the IMS technique and a resolving power of 2.2 × 105 has beenobtained in a recent measurement [ZXS+17].

Similarly to the TOF-Bρ measurements, the ion trajectory in a storagering cannot be measured precisely and hence the mass accuracy is limitedby the velocity dispersion. To overcome this difficulty, a double-TOF-detectorsystem has been constructed at CSRe to correct the momentum dispersionof the stored ions [SXZ+16]. The idea is that by using two TOF detectorsinstalled in a straight section of the storage ring, the velocity of ions canbe determined and thus the information of the actual orbit length can beobtained. In this way, the deviation of the revolution orbit from the centralorbit can be corrected.

Multi-Reflection Time-of-Flight Mass Spectrometry

The newly developed multi-reflection time-of-flight (MR-TOF) technique[WP90] has been widely used to determine the mass of very short-lived nu-clides. The principle is that, by using two symmetric electrostatic mirror elec-trodes facing each other, the ions can be reflected hundreds or thousands oftimes and hence the resolving power can be largely increased. A large numberof ions can be trapped simultaneously and the number of reflection of an ioncan be adjusted easily according to the half-lives of the ions of interest. Thereflection number is usually chosen to be 1000 at ISOLTRAP, and larger than∼ 150 at RIKEN.

The mass resolving power for the MR-TOF mass spectrometry can bedefined as:

R =m

∆m=

t

2∆t, (2.7)

where t is the time of flight and ∆t is the peak width (FWHM). From thedefinition, we known that the resolving power R scales with the time of flightand hence the number of reflection, given the time resolution is constant.

A resolving power of 200,000 (for A = 40 at t = 30 ms) has been reached atISOLTRAP [WWA+13], 600,000 (for A = 133 at t = 49 ms) at GSI [DPB+15]and 203,000 (for A = 12 at t ≈ 10 ms) at RIKEN [ISW+13]. More detailsabout the MR-TOF mass spectrometry will be given in Chapter 8.

Penning-trap mass spectrometry

Nowadays, Penning-trap mass spectrometry provides the most accurateand precise data in atomic mass measurements. Almost all radioactive beamfacilities around the world use (or plan to use) Penning-trap mass measurementsystems [BDN13]. For stable nuclides, a mass precision of an unprecedented

32


level (7× 10−12) [RTP04] has been reached, and for short-lived nuclides, pre-cisions better than 10−7 are routinely obtained. Such high precision allow toperform studies in different fields of physics [BDN13], e.g., in astrophysicalphysics (the study of the composition of neutron-star crusts [WBB+13b] andwaiting points in rapid proton capture process [RKA+04, TXW+11, CSS+04])and neutrino physics [EBB+15].

The principle of Penning-trap mass spectrometry is to measure the cy-clotron frequency ωc of an ion in the magnetic field B, which is related to itsmass-to-charge ratio:

ωc =q

mB. (2.8)

Since the strength of B cannot be known precisely, the measurement of awell-known mass is essential to make a reliable calibration.

From the derivative of Eq. 2.8 with respect to m:

dωcdm

= −qBm2

= −ωcm, (2.9)

the resolving power of Penning traps can be defined:

R =m

∆m=

ωc∆ωc

=νc

∆νc(2.10)

where ωc = 2πνc. The line width ∆νc with which the cyclotron frequency isdetermined is given by [Bol01]:

∆νc ≈ 0.8/Tobs, (2.11)

where Tobs is the observation time of the ion motion in seconds.Thus, the resolving power can be written:

R ≈ 1.25 · νc · Tobs. (2.12)

In order to have a high resolving power, high cyclotron frequency caused byhighly charged state or strong magnetic field, and long observation time aredesirable. The cyclotron frequency of a singly charged ion with mass 100 u inan 8-T field is around 1.2 MHz. A resolving power of 1.5×106 can be obtainedif the observation time Tobs = 1 s.

In this chapter, the basic concept of different types of methods in atomic-mass measurements is introduced. The Penning-trap mass spectrometry willbe detailed in Chapter 7 and the MR-TOF technique will be discussed in depthin Chapter 8.

33

2.3. DIRECT METHODS

34

Chapter 3

The Evaluation Procedures

3.1 General remark

The evaluation of atomic masses is subject to a special way of treatingdata. The present knowledge of atomic masses, as discussed in Chapter 2, canbe categorized into four classes: a) beta-decay energies, b) disintegration en-ergies from light-particle emissions, such as α and proton decays, c) energyreleased in nuclear reactions, and d) mass-spectrometric data (calorimeterscan also yield energy information by detecting heat). All mass measurementsare relative measurements, meaning that every single experimental datum ba-sically establishes a relation between two, or sometimes more, masses, eventhe “absolute” mass measurements compare an ion of interest with carbonclusters [BBH+02]. Obtaining the best value for masses from numerous data,which are sometimes conflicting, is not an easy task. The ideal situation wouldbe that the whole dataset forms a homogeneous system, meaning that all thedata is subjected to the same analysis procedure and errors are given in aunique way. It is however far from realistic. Not all the experimentalists usethe same methods to treat data and even the assigned errors have differentinterpretations. There thus exists no unique way of treating such an inhomo-geneous set of data. One may try to scrutinize, select, correct, or even rejectdata according to some rules to make it as homogeneous as possible. But itdepends strongly on evaluators’ experience.

Another characteristic in mass measurements lies in the fact that a nu-clide can be involved in both nuclear reaction and mass-spectrometric mea-surements, meaning that the number of measurements is inevitably larger thanthat of the involved masses. How to handle such interconnected links is anotherchallenge.

We should also pay attention that input data are not always indepen-dent. For example, the decay energy for a ground-state nuclide can be derivedmore accurately by combining transition to some excited states, while theground-state transition assignment can be based on mass-spectrometric data.

35

3.2. INTRODUCTION TO LEAST-SQUARES METHOD

The evaluation strategy is to consider all the input data as independent. Suchtreatment is of course not perfect but we notice that the correlation betweeninput data affects only some of the ultra-precise measurements.

There are two fundamental hypotheses on which the mass evaluation arebased [Bos77]: a) the mass of an atom is constant and b) masses are additive.The first statement is the key in the least-squares adjustment. The secondone means that if the mass difference between A and B is q1 and between Band C is q2, then the mass difference between A and C is q1 + q2. These twohypotheses are so fundamental that everyone takes them for granted.

3.2 Introduction to least-squares methodSuppose that we have N measurements connecting n masses (N > n).

Each experiment establishes a relation between several nuclides:

qi =∑µ

kiµmµ, (3.1)

where qi denotes the result of the i-th experiment and mµ is the µ-th relatedmass. K = (kiµ) indicates the coefficient matrix and the index i ranges from1 to N and index µ ranges from 1 to n. As mentioned above, the number ofobservations N is usually much larger than that of masses n. It is inadmissibleto discard N − n data and calculate the unknown quantities through mµ =∑

i κµiqi (with∑

i κµikiγ = δµγ), which would results in(N

n

)different values for each mass. This predicament can be solved by using least-squares method [Wap60].

First, we can rewrite Eq. 3.1 using matrix notation:

q = Km, (3.2)

where q={qi} and m= {mµ} are both column vectors. K is an (N × n) di-mensional matrix. The variance σi of the i-th measurement qi is used to builda weight matrix W. If the input data is independent, which is always the case,then W is a diagonal matrix and can be written:

W = diag{1/σ2i }. (3.3)

The normal matrix A can be constructed [rB96]:

A = KtWK, (3.4)

36

CHAPTER 3. THE EVALUATION PROCEDURES

where Kt is the transpose matrix of K. A is an (n×n) square positive definitematrix and its inverse matrix A−1 has the same properties. The whole systemcan be solved by the so-called normal equation:

Am = KtWq, (3.5)

where m is a column vector which contains the adjusted values for the masses.Since A is invertible, the best values for the masses can be obtained by solving:

m = A−1KtWq, or m = Rq. (3.6)

The (n×N) matrix R is called the response matrix. The inverse matrix A−1

is none other than the covariance matrix, with its diagonal element (A−1)µµstanding for the variance of the µ-th mass and its off-diagonal element (A−1)µνsignifying the covariance between two masses mµ and mν .

We can replace the unknown quantity m in Eq. 3.2 by the best value mthen we obtain the adjusted values for each input datum:

q = Km, (3.7)

with its variance:σ2(q) = KA−1Kt. (3.8)

The root of the variance represents the uncertainty of each adjusted inputdatum.

To be general, any linear combination of the adjusted values can be writ-ten as:

q∗ = gtm, (3.9)

where gt denotes a row vector of coefficients. With the help of the covariancematrix, one can calculate the variance of any linear combination of masses:

σ2(q∗) = gtA−1g. (3.10)

Table 3.1: Typical input in Eq. 3.2 and corresponding mass equation.

i-th datum Types∑

µ kiµmµ

1 33P(β−)33S 33P − 33S2 214Th(α)210Ra 214Th − 4He − 210Ra3 19Na(p)18Ne 19Na − 1H − 18Ne4 27Al(p,n)27Si 27Al + 1H − 1n − 27Si5 C7H14 − 98Ru 7·12C + 14·1H − 98Ru6 Penning Trap M − k ·Mref

Table 3.1 lists some typical input data. One may already see that thecoefficient matrix K in Eq. 3.2 is a sparse matrix: only a few nuclides are

37

3.3. REDUCTION OF THE PROBLEM

involved in each measurement, making most of the elements null in K. Fordecay and reaction measurements, k = 1 for entrance channel and k = −1 forexit channel. For mass-spectrometric measurements k is not always an integernumber (see Section 4.1 for discussions).

Why least-squares?One may ask why use the least-squares method to extract the best val-

ues from a wealth of observations. I think there are mainly two reasons forthat. First, the mass itself is usually not the quantity of interest ∗ and whatmatters most of the time is a certain linear combination of masses such asseparation energy, beta-decay energy, etc. However, the precision of any com-bination in mass could not be obtained without considering the error matrix,which appears automatically in the least-squares method. Secondly, when theleast-squares method is carried through over the whole data set, one can judgethe correctness of an input datum by comparing with the adjusted one. Suchtreatment could reveal undiscovered systematic errors among some measure-ments. The adjustment is justified only if the consistency of the input data issatisfied.

3.3 Reduction of the problemPrimary and Secondary

Before performing the least-squares adjustment, all the data is separatedinto two groups. Suppose we have a series of measurements like in the firstcolumn below:

A−B = q1

A− C = q2

B − C = q3

B −D = q4

C −D = q5

D − E = q6

E − F = q7

E −G = q8

H − I = q9

=⇒

A−B = q1

A− C = q2

B − C = q3

B −D = q4

C −D = q5

D − E = q6

=⇒

A−B = q1

A− C = q2

B − C = q3

B −D = q4

C −D = q5

Each capital letter represents the mass of a nuclide and qi represents the cor-responding i-th Q-value. As the nuclides H, I, G, and F appear only once in

∗. One exception would be the Avogadro Project of redefining the kilogram, where theatomic mass of 28Si would be used as an input parameter.

38


the system, they can be removed temporarily with their corresponding inputdata. What remained are listed in the middle column. E should be also re-moved since E − F and E −G were removed in the first round. Finally, onlyfive equations which connect four unknown masses are left in the last columnand they will be entered into the least-squares evaluation.

The nuclides that are left are primary and their related equations areprimary data, while the nuclides that are removed are secondary nuclides andthe related equations are called secondary data. This classification does notmean that one is more important than the other: the primary data are sub-ject to the least-squares process to obtain the masses of primary nuclides whilesecondary data give information for a specific nuclide and will never improvethe precision of other nuclides (secondary data has no contribution to the χ2

(detailed next). Figure 3.1 illustrates this example. The masses of A,B,C,Dform an overdetermined system and they are subject to the least-squares ad-justment. The masses of secondary nuclides can be derived, without any loss ofinformation, in a straightforward way once the masses of the primary nuclidesare fixed. Nuclides, H and I, which have no connection to neither primarynuclides nor secondary nuclides are called unconnected nuclides. Their massescan be obtained only if a link is built which connects them to an experimen-tally known nuclide (see Chapter 6). The link is estimated and Ame assigns aspecial symbol “#” for those estimated masses.

In Ame, we assign a degree to each nuclide: all primary nuclides havedegree 1 while secondary nuclides have their degree increasing from two tohigher values depending on how far they are from the primary nuclides. Thedivision of the input data into two groups was adopted for two reasons: a) inthe old times, the computation ability was strongly limited by the dimension ofthe normal matrix in Eq. 3.4. The time necessary for the inversion of a matrixis the cube of the number of the primary nuclides (in 1960, the inversion of amatrix with 670 rows needed a CPU machine time of some 20 days [Kön60]);b) the separation of secondaries allows to localize abnormalities more easily.And also allows to check the dependencies faster. Even though the calculationtime nowadays is no longer a problem for Ame (it takes only 30 s to completethe inversion), we still keep such a policy in our evaluation process, since itis much more pleasant to get a result almost 30 times faster (assuming thenumbers of primary and secondary nuclides are 1000 and 2000, respectively).

Fig. 3.2 displays the connection plot in the 163Ho region. Each symbolrepresents a nuclide and each line represent an experimental datum. The di-rect measurement of the mass difference between 163Ho and 163Dy is essentialto address the neutrino mass [EBB+15]. The two nuclides of interest 163Ho and163Dy, which are colored in red, are connected directly by mass spectrometry(double line) and EC-decay spectrometry (single line). They also connect indi-rectly through the neighboring nuclides through several different paths. Onlythe Ame procedure can disentangle such a complex connection and obtain areliable Q-value for the mass difference between 163Ho and 163Dy.

39

3.3. REDUCTION OF THE PROBLEM

A B

C D E

F

G

H

Iy5

y1

y2 y4

y3

y6

y7

y8

y9

Figure 3.1: Connection plot with primary, secondary and unconnected items.

Figure 3.2: Connection plot in the 163Ho region. Each symbol (square andcircle) represents a nuclide: the large ones denote nuclides that would be usedin the least-squares procedure; the small ones denote secondary nuclides thatwould not be used in the least-squares procedure. The upper red square symbolindicates 163Dy and the lower one indicates 163Ho.

40


Pre-averaging

Two or more experiments measuring the same quantity can be averagedwithout losing information. The adjusted mass values will not be affectedwhether such pre-averaging is applied or not, provided that the internal errorsare assigned to the weighted means. Our policy is to give only one value for aspecific reaction or decay Q-value. Such pre-averaged data are considered asparallel data and their weighted mean is calculated by:

q =

∑i qiσ

−2i∑

i σ−2i

, (3.11)

where σi is the uncertainty of the i-th Q-value and its associated internal erroris:

εint = (∑i

σ−2i )−1. (3.12)

However, one cannot use blindly the internal error as the adjusted inputerror since it makes no sense to enter some data which conflicts with eachother. Under such circumstance, besides the internal error, one also needs tocalculate the external error:

εext =

∑i(q − qi)2σ−2

i

(N − 1)∑

i σ−2i

, (3.13)

where N is the number of the averaged items. If the ratio χn = εext/εint† is

larger than 2.5, then the internal error will be replaced by the external error.The pre-averaging procedure is also applied to parallel data, i.e., the data

connecting the two of the same nuclides. The masses of some light particlesinvolved in reactions can be regarded as constants because they are mea-sured with very high precision by other experiments. That is to say that somereaction data that gives the mass difference between two identical nuclidescan also be averaged. For example, two equations 48Ca(p,g)49Sc=9628.7 ±3.6 keV [VCB68] and 48Ca(d,n)49Sc=7404 ± 4 keV [GMDN68] give the sameinformation for the mass difference between 48Ca and 49Sc because the massesof proton, neutron, and deuteron are well known (zero mass for rest photon).Then the above two equations can be rewritten as q1 = 9628.7−M(1H)± 3.6and q2 = 7404 − M(2H) − M(n) ± 7, where both q1 and q2 represent themass difference M(48Ca) − M(49Sc). Inserting the light masses, one obtainsq1 = 2339.7 ± 3.6 and q2 = 2340 ± 7. The average of the two data givesq = 2339.7 ± 3.2 keV with external error 0.05 keV, which shows excellentconsistency of the two data. In AME2016, 2977 data were replaced by 1186averages. As can be seen from Fig. 3.3, 23% of the data have χn beyond unity,1.6% beyond two and none beyond 2.5, meaning a satisfactory treatment forthe pre-averaging data was achieved.

†. This ratio is called Birge Ratio and has exactly the same definition as the consistencyfactor described below as n = 1 in Eq. 3.15

41

3.4. χ2 TEST AND CONSISTENCY FACTOR

0.0 0.5 1.0 1.5 2.0 2.5 3.0

χn

0

10

20

30

40

50

60

70

80

Counts

Figure 3.3: Birge Ratio of all the parallel data

3.4 χ2 test and consistency factorThe prerequisite for the least-square method described above is that each

input datum should have a distribution function with a mean (expected value)and a variance (square of the standard deviation of the mean) such that thevalue of χ2:

χ2 = (q−Km)tW(q−Km) = vtWv =∑

i

Wiv2i , (3.14)

is minimum. Here we introduce a residual vi which is defined as the differencebetween the input value and the adjusted one. The expectation value and thevariance of χ2 are:

χ2 = N − n = f and V (χ2) = 2(N − n) = 2f, (3.15)

where f is the number of degrees of freedom. χ2 serves as an indicator for theconsistency of the input data. One can also define a more intuitive quantitycalled consistency factor:

χn =

√χ2

f, (3.16)

where its expectation value and one standard deviation are:

χn = 1 and σ(χn) = 1/√

2f. (3.17)

One may expect that χn would be very close to unity if the errors areassigned properly for each input datum. It may, however, be rather large.One would not be bothered too much if χn is implausibly smaller than one,which would be due to the fact that the assigned errors are too large and theexperiments were turned out to be more reliable than what we had expected.On the other hand, if the consistency factor is much too large (χn � 1) then

42


we might suspect correctness of the input data, which would suffer from anundiscovered systematic error.

In 1961, the first Ame adjustments [EKMW61] were performed for massdoublets and reaction data separately. The evaluation for the mass-doubletdata taken from the Brookhaven and Minnesota laboratories yielded consis-tency factors of 3.58 (N = 13, n = 3) and 2.35 (N = 39, n = 8) for theformer one, and 2.65 (N = 36, n = 12) for the later. The evaluation of thereaction data yielded a consistency factor of 0.60 (N = 73, n = 19). Basedon the pre-adjusted results, all the reported errors for the mass doublets fromthese two groups were multiplied by the corresponding consistency factor be-fore they were combined with other nuclear and decay measurements in thefinal adjustment.

In principle, if the consistent factor defined in Eq. 3.16 is significantlylarger than one, then all errors used should be multiplied by

√χ2/(N − n) in

order to achieve overall consistency. This is the last thing we want to do sincea large χ2 value could arise from a few number of data or a special group.Correcting, or sometimes discarding, all the inconsistent data which deviateslargely from the least-squares result and re-evaluating the input data withoutthose anomalies could greatly improve the consistency of the whole system.

Another quantity that is also of interest is the Partial ConsistencyFactor (PCF), which is defined for a group of p data:

χpn =

√√√√ N

N − n1

p

p∑i=1

Wiv2i . (3.18)

One can consider two groups of data: the reaction- and decay-energymeasurements on one side, and the mass-spectrometric ones on the other side.One can also consider the Partial Consistency Factor for the data froma given laboratory using a certain device. The advantage of using PartialConsistency Factor is that we do not need to separate the system intoseveral sub-systems ‡ which should have no correlation with each other. ThePartial Consistency Factor has a value one for all the reaction and de-cay data since they have been studied so thoroughly and there is very littledoubt about the correctness of the ground-state assignment: the distance be-tween the ground state and the first exited state is usually so large that afaulty assignment would result in a glaring inconsistency. The determinationof the ground-state masses from mass-spectrometric experiment could mixother isomeric states due to the limited resolving power. At present, Ame ap-plies Partial Consistency Factors of 1.0, 1.5, 2.5, and 4.0 to the mass-

‡. It is not always practical, I would say for most of the time impossible, to obtain asub-group of data from the whole data set, since it requires that the data are treated in ahomogeneous way; moreover, the condition N > n can hardly be met unless one disregardssome other data.

43

3.5. FLOW-OF-INFORMATION MATRIX

spectrometric data, depending on the agreement between the experimentalresults and the adjusted ones.

One should keep in mind that the Partial Consistency Factor can-not be derived from any statistical theory. It only indicates the degree ofconsistency for a set of data among the full adjusted system. Of course thedefinition is such that χpn reduces to χn if the sum is taken over all the inputdata.

3.5 Flow-of-information matrixUp to now we can calculate the best values for all masses and obtain

the adjusted input values using the least-squares method. Besides, we are alsointerested in finding out the information an equation brings to each mass. Theflow-of-information matrix, discovered by G. Audi in 1986 [ADLW86], allowsus to trace back the contribution of each input datum on each mass. Theflow-of-information matrix is defined as:

F = Rt ⊗K, (3.19)

where the symbole ⊗ means the term-by-term product operation of the corre-sponding elements in transpose matrix Rt and that of K. It is very practicalto define F this way, since, as proved in [ADLW86], each matrix element (i, µ)represents the contribution brought by the input datum i in the determinationof the mass mµ. The contribution here is called influence. The sum of all theinfluences in a row of F is the significance of the corresponding datum, whichsignifies the overall contribution brought by this input datum to all the relatedmasses. The significance of each input datum is less or equal to one and woulddecrease as more data comes into the system. The significance defined in thisway is exactly the quantity which can be obtained by squaring the ratio of theadjusted uncertainty over the input one. The sum of a column of F is the totalinfluences brought by all the input data to a specific mass, which is always100 %. The flow-of-information matrix has its name because it sheds light onhow the information of a particular datum “flow” into each mass. It can alsohelp plan future experiments in order to achieve maximum precision.

3.6 Local adjustment in the lightest mass regionThe β-decay Q-value of 3H(β−)3He is very important in the determina-

tion of the neutrino mass. The accurate measurement of the mass differencebetween 3H and 3He is essential for understanding the shape of the tritiumβ-decay spectrum and for investigating the systematic effects in the endpointregion [ABB+11]. However, we noticed that the evaluation outcome in thismass region is not satisfactory. In this section, a detailed example will be

44


given to illustrate how to use the least-squares method to carry out a localevaluation. Fig. 3.4 shows all the related nuclides 1H, 2H, 3H, and 3He under

Table 3.2: Related information for the local adjustment in the lightest massregion.

Refs Penning trap Equation input Q-value (µu)[SBN+08] SMILETRAP 2·1H − 2H 1548.28649(0.00035)[VDZS01] Seattle 12·1H − 12C 93900.3865(0.0017)[ZJ15] Seattle 6·2H − 12C 84610.66834(0.00024)[ZJ15] Seattle 4·3He − 12C 64117.28668(0.00017)

[MWKW15] FSU 3He − 1H − 2H −5897.48771(0.00014)[MWKW15] FSU 3H − 3He 19.95934(0.00007)

consideration, and they are connected by lines which represent the experimen-tal data with corresponding references. Since the number of experimental datais larger than that of nuclides, the least-square method is applicable. Since themass of 12C is defined as mass standard, it is not included in the mass adjust-ment. Though 3H is a secondary nuclide, we also include it in the least-squareprocedure for convenience. In [MWKW15], three frequency ratios, i.e. 3H/HD,3He/HD, and 3H/3He, were given while only two of them were considered asindependent. The less precise datum which connects 3H with 1H and 2H wasnot used in AME2016. We will see that the inclusion of this secondary datumhas no effect on other masses. What’s more, the evaluation is “local”, whichmeans that they are isolated from the rest of the chart. The mass of 2H isalso determined by other experiments. Here, we concentrate only on the mostrelevant data for simple illustration.

All input data for the evaluation is listed in Table 3.2. The first columnlists the publication references, followed by the corresponding Penning-trapspectrometer in the second column, the third column lists the equations en-tered in the adjustment and the fourth column is the related Q-value in µu.Once we have all the equations, we can build the coefficient matrix K:

K =

1H 2H 3H 3He

2 −1 0 012 0 0 00 6 0 00 0 0 4−1 −1 0 10 0 1 −1

(3.20)

and its transposed matrix Kt :

45

3.6. LOCAL ADJUSTMENT IN THE LIGHTEST MASS REGION

Figure 3.4: Connection plot of the lightest nuclides. Each line represents anexperimental datum. The corresponding reference papers are also indicated.

Kt =

2 12 0 0 −1 0−1 0 6 0 −1 00 0 0 0 0 10 0 0 4 1 −1

(3.21)

The weight matrix W can be constructed from Eq. 3.3 using the uncer-tainties listed in the third column in Table 3.2:

W =

8163265.306 0 0 0 0 00 346020.7612 0 0 0 00 0 17361111.11 0 0 00 0 0 34602076.12 0 00 0 0 0 51020408.16 00 0 0 0 0 204081632.7

(3.22)

Hence, the normal matrix A can be constructed:

A = KtWK =

133500459 34693877.55 0 −51020408.16

34693877.55 684183673.5 0 −51020408.160 0 204081632.7 −204081632.7

−51020408.16 −51020408.16 −204081632.7 808735258.8

(3.23)

46


The inverted matrix A−1 is:

A−1 =

7.82246E-09 −3.49643E-10 6.30553E-10 6.30553E-10−3.49643E-10 1.48648E-09 9.59257E-11 9.59257E-116.30553E-10 9.59257E-11 6.61514E-09 1.71514E-096.30553E-10 9.59257E-11 1.71514E-09 1.71514E-09

(3.24)

The response matrix R is:

R = A−1KtW =

0.130567808 0.032480788 −0.036421186 0.087273731 −0.349094922 0−0.017842986 −0.001451806 0.154841561 0.013276912 −0.053107649 00.009511671 0.002618212 0.009992259 0.237389529 0.050441885 10.009511671 0.002618212 0.009992259 0.237389529 0.050441885 0

(3.25)

Combining with the vector q which contains the experimental results:

q =

1548.2864993900.386584610.6683464117.28668−5897.48771

19.95934

(3.26)

The evaluated values of all nuclides can be obtained:

m = Rq =

7825.03191614101.77800716049.28105616029.321716

(3.27)

and their uncertainty can be obtained from the root of the diagonal elementsof A−1. Finally, the masses and uncertainties for each nuclide are :

M(1H) = 7825.031916± 0.000088 µuM(2H) = 14101.778007± 0.000039 µuM(3H) = 16049.281056± 0.000081 µuM(3He) = 16029.321716± 0.000041 µu

(3.28)

The flow-of-information matrix is:

F = Rt ⊗K =

0.261135616 0.017842986 0 00.389769462 0 0 0

0 0.929049366 0 00 0 0 0.949558115

0.349094922 0.053107649 0 0.0504418850 0 1 0

(3.29)

From the flow-of-information matrix in Eq. 3.29, we can acquire theinformation of the importance of each input in the evaluation. For example,

47

3.7. REMOVAL OF CERTAIN INPUT DATA

from the first row of F we know that the influences of [SBN+08] are 26.11%and 1.78% in the determination of the masses of 1H and 2H, respectively, whichresult in the significance of 27.89%. From the last row of F, we can see thatthe influence of [MWKW15] is 100% in the determination of the mass of 3Hand 0% elsewhere, which guarantees that the secondary datum does not affectother primary nuclides.

Table 3.3 lists the results of the adjusted Q-values, and v/s indicatesthe difference between the input Q-value and adjusted Q-value divided bythe uncertainty of the input Q-value. The largest deviation comes from themass measurement of 3He/HD [MWKW15], which results in v/s of 3.6. Theconsistency factor χn is 3.4 (N = 6, n = 4) calculated from Eq. 3.16, whichindicates that the measurement is subject to systematic errors which have notbeen found.

Table 3.3: Adjusted Q-values in the lightest mass region. v/s indicate thedifference between the input Q-value and the adjusted Q-value divided by theuncertainty of the input Q-value.

Refs Equation input Q-value (µu) adjusted Q-value (µu) v/s[SBN+08] 2·1H − 2H 1548.28649(0.00035) 1548.28583( 0.00019) 1.9[VDZS01] 12·1H − 12C 93900.3865(0.0017) 93900.38300(0.00106) 2.1[ZJ15] 6·2H − 12C 84610.66834(0.00024) 84610.66804(0.00023) 1.2[ZJ15] 4·3He − 12C 64117.28668(0.00017) 64117.28686(0.00017) −1.1

[MWKW15] 3He − 1H − 2H −5897.48771(0.00014) -5897.48821(0.00009) 3.6[MWKW15] 3H − 3He 19.95934(0.00007) 19.95934(0.00007) 0

If one uses the most recent result of the proton mass from [HKLR+17],which is over four times more precise than that of [VDZS01], the consis-tency factor will decrease to 3.0. However, the inconsistency still remains.In AME2016, the results from [ZJ15] were not used, since the mass differencebetween the ions of interest and reference ions (carbon) are relatively large.This case has been discussed in detail in section 7.1 in AME2016 [HAW+17]).

Recently, the authors in [MWKW15] remeasured the frequency ratio of3He to HD+, which yields the mass difference of −5897.48780(0.00007) µu[HSFM17]. The new result is in good agreement with their previous result andthus validates our choice in AME2016.

3.7 Removal of certain input dataAme deals with all kinds of input data that are related to atomic masses.

We apply a special label to each of them to signify its role in the mass evalu-ation. All cases discussed here are not used.

Some experiments would present the same input item with different pre-cision. If one is three times less precise than the other, or if its value can be

48


deduced by the combination of other data, it is marked with label “U” (forunweighted).

In some cases, discrepancies exist between two experiments which givethe same information, or a single input datum has a strong conflict with theprevious adjustment compared with its reported precision. By looking into theoriginal paper, one may find a reason to doubt their reported value and suchdata would be marked with “F”.

Sometimes, interesting results appear in the abstract of conference pro-ceedings or annual reports but no more information could be obtained. Suchresults would be marked with “C” if they were not compatible with otherexperimental results.

The most worrisome cases would be those whose results have been pub-lished in regular journals in which no fault could be found. Including theseresults would cause a large contribution to χ2, which could give some impres-sion that the evaluation is wrong. In this case, they would be marked with“B”.

An old result would be replaced by a new one from the same laboratorywith higher precision. In this case, only the latest result would be adopted andthe old one would be marked with “O”.

Including some experimental data could violate the continuity of the masssurface (see Chapter 6). Our policy is to replace them by the estimates fromthe trends of the mass surface if the deviation is larger than 200 keV. Suchreplaced data would be marked with “D”.

Some nuclides involved in the reaction or decay measurements could bealso measured directly by mass spectrometry. Using both data would makethese nuclides primary. We can, without loss of any information, replace oneof them by an equivalent expressed like the other one. The two results willbe then pre-averaged inside the adjustment program. We avoid using all thesedata since they would only effect in a local region. We would cut one of theoriginal links and replace it with another input item with the same precision.Such replaced data would be marked with “R”.

In this chapter, the philosophy of Ame in treating experimental datais described. The most important feature of Ame is to consider the massesas parameters in the least-squares method. Since the masses enter linearlyin each input equation, the parameters (masses) can be solved without ap-proximation. Up to now, ten atomic-mass tables have been published basedon this method: AME1955 [Wap54a, Wap54b, Hui54], AME1961 [EKMW60a,EKMW60b], AME1964 [MTW65], AME1971 [WG71], AME1977 [WKB77],AME1983 [WA85], AME1993 [AW93], AME2003 [AWT03], AME2012 [WAW+12],and AME2016 [WAK+17].

Next, the developments for the latest Ame adjustment AME2016 will beillustrated.

49

3.7. REMOVAL OF CERTAIN INPUT DATA

50

Chapter 4

Developments for AME2016

4.1 Calculation of molecular binding energies forthe most precise mass measurements

The most precise mass-spectrometric measurements use Penning-trap massspectrometry (see Chapter 7). One can measure not only the mass of a singleatomic ion, but also the mass of a molecular ion in experiments [DNB+95].For most molecules, the binding energy represents typically a correction of afew parts in 1010, and its uncertainty only limits the accuracy of the neutralatomic mass to a few parts in 1012 [RTP04]. For measurements with pre-cision not better than 10−9 (100 eV/100 u), the molecular binding energycould be neglected without losing too much accuracy. For precisions exceed-ing 10−10 (10 eV/100 u), which are routinely attained at MIT [DNBP94] andFSU [MWKW15], the importance of the molecular binding energy should betaken into account so as to obtain accurate atomic masses.

Reducing frequencies to linear equations

The measured quantity in a Penning trap is a frequency ratio. As men-tioned in Chapter 3, the input data should be linear in mass so that theleast-squares method can treat such an input item without approximation.Ptrap (see Appendix C.1 in [AWW+12]) is a program of deriving the massdifference from Penning-trap mass measurements carried out with atoms. Therelation between the ion of interest and the reference ion can be expressed as[AWW+12]:

M −meq +B

Mr −meqr +Br

qrq

= R, (4.1)

where R is the frequency ratio between the ion of interest and the referenceion (in the following, quantities with subscript r represent the reference ionand without subscript represents the ion of interest), M is the total atomic

51

4.1. MOLECULAR BINDING ENERGY

mass (in atomic mass unit “u”), q is the number of electrons removed from theatom, me is the electron mass, and B is the ionization energy of the atom.

If molecules are involved in experiments, the molecular binding energy Dshould be included in the calculation. In this case, Eq. 4.1 should be rewritten:

M −D −meq +B

Mr −Dr −meqr +Br

qrq

= R. (4.2)

The total atomic mass M can be expressed as M = M +A, where M isthe mass excess and A is the mass number. Eq. 4.2 thus becomes:

A+M −D −meq +B = Rq

qr(Ar +Mr −Dr −meqr +Br)

M −R q

qrMr = meq(1−R) + Ar(

q

qrR− A

Ar) +R

q

qr(Br −Dr)− (B −D),

(4.3)which comprises all necessary components to calculate the mass M .

Since the experimental data should be entered as linear equations in mass,we define a constant C as a three-digit decimal approximation of A over Arand want to find a quantity y such that:

y = M − CMr. (4.4)

The advantage of constructing such a quantity is that, once the precisionof the reference mass Mr is improved, the mass M can be recalculated auto-matically. Combining Eq. 4.3 and Eq. 4.4, one finds the quantity y has theform:

y = Mr(Rq

qr− C) +meq(1−R) + Ar(

q

qrR− A

Ar) +R

q

qr(Br −Dr)− (B −D)

= y1 + y2 + y3 + y4.(4.5)

Mr is generally smaller than 0.1 u, me is of the order of 500 µu, (R qqr−C) and

(1 − R) are a few 10−4, Ar=100 u for mass number 100, R − AAr

varies from10−6 to 10−4, Br−Dr has order of tens of nu (see below). In this case, Eq. 4.4can be entered directly into the Ame least-squares adjustment.

The uncertainty on y can be expressed from the variation dy:

dy = dy1 + dy2 + dy3 + dy4, (4.6)

where

52

CHAPTER 4. DEVELOPMENTS FOR AME2016

dy1 =q

qrMrdR + (R

q

qr− C)dMr ≈ dR ∗ 105 µu

dy2 = −meqdR ≈ −dR ∗ 5 ∗ 102 µu

dy3 = Arq

qrdR ≈ dR ∗ 108 µu

dy4 =q

qr(Br −Dr)dR ≈ dR ∗ 10−2 µu.

(4.7)

dy1, dy2 and dy4 are always negligible compared to the third term. Only the 3rdterm dy3 contributes to the final uncertainty. Although the quantity y dependson the reference mass, as can be seen from y1 in Eq. 4.5, the multiplicationof Mr by the factor (R q

qr− c), which has an magnitude of 10−4, practically

removes any dependence of the measured mass on the reference mass.

Bond Dissociation Energy

The bond dissociation energy (BDE) is a quantity which signifies thestrength of a chemical bond and has its synonym of molecular binding energy(D). The bond dissociation energy Do for a bond A-B which is broken througha reaction:

AB → A+B

is defined as the standard enthalpy change at a specified temperature [Dar70]:

Do(AB) = ∆Hf o0 (A) + ∆Hf o0 (B)−∆Hf o0 (AB), (4.8)

where ∆Hf o0 is the standard heat of formation and its value for atoms andcompounds is available on the NIST Chemistry Webbook [LM16a]. The su-perscript in Do refer to the gaseous state at 0 K temperature.

Unlike diatomic molecules which involve only one bond, polyatomic moleculeshave several bonds and their BDEs are the sum of all the single bonds. Forexample, if one wants to know the BDE of CH4, one needs to calculate theBDE of CH3−H, CH2−H, CH−H and C−H:

Do(CH3 −H) = ∆Hf o0 (CH3) + ∆Hf o0 (H)−∆Hf o0 (CH4)

Do(CH2 −H) = ∆Hf o0 (CH2) + ∆Hf o0 (H)−∆Hf o0 (CH3)

Do(CH −H) = ∆Hf o0 (CH) + ∆Hf o0 (H)−∆Hf o0 (CH2)

Do(C −H) = ∆Hf o0 (C) + ∆Hf o0 (H)−∆Hf o0 (CH)

(4.9)

Summing over all four bonds above, we can obtain the BDE for CH4:

Do(CH4) = D(CH3 −H) +D(CH2 −H) +D(CH −H) +D(C −H)

= ∆Hf o0 (C) + 4 ·∆Hf o0 (H)−∆Hf o0 (CH4)

53

4.1. MOLECULAR BINDING ENERGY

For a polyatomic molecule which has the form AnBkCi, its BDE can begeneralized:

Do(AnBkCi) = n ·∆Hf o0 (A) + k ·∆Hf o0 (B) + i ·∆Hf o0 (C)−∆Hf o0 (AnBkCi)(4.10)

Below two examples of the calculations of molecular binding energies willbe given.

Atomic Masses of Tritium and Helium-3

By measuring the cyclotron frequency ratios of 3He+ and T+ to HD+,using HD+ as a mass reference, atomic masses for 3He and T were obtained[MWKW15]. The essential part here is to calculate the molecular bindingenergy Do(HD). Applying Eq. 4.8, the molecular binding energy of HD canbe obtained:

Do(HD) = ∆Hf o0 (H) + ∆Hf o0 (D)−∆Hf o0 (HD)

= 218 + 221.72− 0.32 kJ/mol

= 439.4 kJ/mol

(4.11)

or alternately,Do(HD) = 4.55 eV (using conversion factor 1 eV = 96.485 kJ/mol).Combining the ionization energies B(HD)=15.44 eV, B(3He)=24.59 eV andB(T)=13.60 eV from [KRR16] and atomic masses of H and D from AME12,the mass differences and their uncertainties can be obtained in Table 4.1.

Table 4.1: Frequency ratios and mass differences.

Ion pair Frequency ratio Mass difference Value (µu)HD+/3He+ 0.998048085153(48) 3He − H − D −5897.48771(14)HD+/T+ 0.998054687288(48) T − H − D −5877.52837(14)

The correction of the molecular binding energy Do(HD) = 4.55 eV is30 times larger than the uncertainty of the mass differences, which is 0.14 nu≈ 0.14 eV (Table 4.1).

13C2H+2 and 14N+

2 mass doublet

In the work of [RTP04] at MIT, a cyclotron frequency ratio R = 0.999421460888(7)of relative precision of 7× 10−12 has been obtained for molecular ions 13C2H+

2

and 14N+2 . We first calculate Do(C2H2) by applying Eq. 4.10:

54


Do(C2H2) = 2 ·∆Hf o0 (C) + 2 ·∆Hf o0 (H)−∆Hf o0 (C2H2)

= 2 · 716.7 + 2 · 218.0− 227.4

= 1642.0 kJ/mol= 17.0 eV

Do(N2) = 944.9 kJ/mol = 9.793 eV is imported from Ref [LK12], since thevalue ∆Hf o0 (N2) is not on the list of [LM16a]. Combining the ionization energyB(C2H2) = 11.4 eV and B(N2) = 15.6 eV, we obtain the mass difference:

13C− H− 14N = 8105.86300(10) µu.

In the original paper [RTP04], this equation is given as:

13C− H− 14N = 8105.86288(10) µu,

which differs by 0.12(10) nu from the new calculation, because an updatedtabulation of molecular binding energies [LM16a, LK12] is used.

From the two examples discussed here, we notice that the consideration ofmolecular binding energies is essential to obtain correct input values for precisemass-spectrometric data. Even using the updated stand heat of formationwill change the original result significantly. In AME2016, all precise mass-spectrometric data has been recalculated, based on the latest standard heatof formation [LM16a].

4.2 Experimental α-decay and proton-decay en-ergies

According to AME2016, more than 65% of the input data in the massrange A > 200 results from α-decay experiments. In lighter mass regions thereare also a large number of proton-decay data, which shares many similaritieswith α-decay data . Energies from α and proton decay yield information ofcapital importance for deriving mass values of superheavy and exotic nuclides.But when we refer to α-decay or proton-decay experiments, we often find someconfusion: energy values referred to by one author as the particle kinetic energyE could sometimes be referred to as the decay energy Q by another.

The decay energy is the sum of kinetic energies of the emitted particleand the recoiling daughter nuclide. Typically, in an α-decay measurement, theα particle carries about 97-98% of its Q value and the recoiling nuclide carriesthe rest. In the literature one can find too many cases of confusion, especially inthe case of proton-decay experiments where Qp and Ep are numerically closerto each other. Sometimes, the confusion could be solved through a meticulous

55

4.2. α- AND PROTON-DECAY ENERGIES

inspection of the paper and a discussion with the authors. However, it remainsunclear in a few other cases.

There are four major experimental approaches for α-decay measurements:The first one uses a magnetic spectrograph [GR71], from which the α kineticenergies are determined by direct measurements of the orbit diameters and themagnetic induction field. All α-energy standards have been measured usingthis method. The second one uses the scintillating bolometer technique, whichdetects the total α-decay energy at temperatures below 100 mK [DMCD+03].In third method the nuclide of interest is implanted into a foil and the α particleis detected by the surrounding Si detectors [AEH+10]. Last but not least, theradioactive species, which are produced in a nuclear reaction, are directlyimplanted into a Si detector (a double-sided silicon-strip detector (DSSD) ora resistive-strip detector [Kno12]). The first three methods measure either thepure α-particle energy or the total α-decay energy, while the last implantationmethod detects the α (or a proton in proton decay) particle and the heavyrecoil daughter nuclide in coincidence. This method has been widely used inrecent years and will be discussed in detail.

In order to perform accurate decay experiments, detectors should be cal-ibrated carefully. If the calibration is not applied correctly, then an erroneouspublished value could follow, sometimes with a deviation not negligible com-pared to the claimed precision.

Particle energy vs. Decay energy

Magnetic spectrographs have been used for absolute α-particle energymeasurement [GR71, RWK86]. Since it is the measurement of a single α-particle movement in spectrometer, one obtains α-particle energy Eα. For theradioactive source measurement, where the emitted particle has been detectedin the Si detector while the recoiling nuclide remained in the sample, oneobtains also purely the α-particle energy (except for a small correction dueto dead-layer absorption). These two types of experiments clearly establish aparticle energy Eα from which decay energy Qα can be kinematically derivedfrom classical mechanics:

Qα =M

M −MHeEα, (4.12)

where M and MHe-4 are the atomic masses of the parent and of the helium-4atoms, respectively (see Eq. 4.21 for cases requiring relativistic calculation).In order to obtain the proton-decay energy Qp from proton-energy Ep, oneshould replace Eα and MHe-4 by Ep and MH (hydrogen mass), respectively.

Cases would be less straightforward for measurements performed by im-plantation methods, where not only the emitted particle but also a partialenergy of the recoil are measured at the same time.

56


Energy calibration

The position-sentitive silicon strip detector (DSSD) is one of the mostwidely used detectors in decay-spectroscopy experiments not only for its highenergy resolution, but also for its ability to record the position of implantationof the particle which is necessary to identify decay chains (thus the sensitivitycan be enhanced significantly by reducing the background).

Suppose a simple case where there are three equidistant lines in an α-decay spectrum in Fig. 4.1. Two well-known α-energy activities line-1 (withE(α1) = 5000 keV) and line-2 (with E(α2) = 5200 keV) are used as calibrantsand line-3 is assigned to the unknown nuclide. If the detector does not detectthe recoiling nuclide in experiments as in Fig. 4.1 (a), then what is measuredwould be the α-particle energy and E(α3) = 5400 keV is easily obtained. In thethe opposite case where the detector measures all the energy of the recoilingion, then the energy scale will change to Fig. 4.1 (b). If line-1 and line-2correspond to nuclides with mass number A = 150, the new scales will changeto Qα(line-1) = 5137 keV and Qα(line-2) = 5342 keV based on Eq. 4.12. In thiscase we measure the α-decay energy Qα and obtain Qα(line-3) = 5547 keV.

Figure 4.1: Illustration of α-decay spectra where line-1 and line-2 are calibrantsand line-3 is unknown at an equal distance from line-2. (a) Case for which thedetector detects only the α-particle energy. (b) Case where the detector detectsalso the recoiling nuclide.

If line-3 corresponds to a nuclide with mass numberA = 150, its energy Eαis deduced to be 5399 keV according to the transformation of Eq. 4.12, whichis 1 keV smaller than the value obtained from Fig. 4.1 (a). However, if line-3corresponds to a nuclide with a different mass number, for example, A = 200,Eα will increase from 5400 keV to 5436 keV, which is off by 36 keV. Thedeviation would become bigger for super-heavy nuclide α-decay measurementsin case the mass of calibrants is much smaller than that of α-decaying daughter.

57


Figure 4.2: Detection efficiency K for different species at different recoilingenergy ER in Si-detector. The range of ER selected here covers most of thedecay experiment cases.

In real case, the detector is not 100% sensitive to the recoiling nuclide. In thenext section, I will show how to take into account the efficiency of the detector.

Detection efficiency for the heavy recoil

The recoiling ions lose their energies in the Si detector in two ways: ex-citation and ionization of the electrons of the atoms (electronic process), orcollision with nuclei of the atoms (nuclear process). The electronic processproduces a signal in the detector, while the nuclear process does not. Knowl-edge of both processes is important for implantation α-decay and proton-decayexperiments where the heavy recoil is detected simultaneously with the lightparticle. In 1963, Lindhard et al. [LNST63] derived a theory to describe theseprocesses, from which the detection efficiency K was defined as:

K =ηRER

=kg(ε)

1 + kg(ε), (4.13)

where ηR is the part of the recoiling energy that is effectively detected in thedetector, ER is the total recoiling energy, ε is called the “dimensionless reducedenergy" which is related to ER, k is a coefficient related to the mass numberand the atomic number of the recoil nuclide and the target nuclide, g(ε) is asemi-empirical function (for more details refer to [LNST63]). This theory wasderived to predict the detected energy of heavy atomic projectiles in matterand agrees well with experimental data [Rat75, HMV+82].

Fig. 4.2 shows the calculations of the detection efficiency K for different

58


nuclides [HA17] based on Lindhard’s theory. For light nuclides (e.g. 20Ne and40Ca), the detection efficiencies increase rapidly as their energies increase. Forintermediate mass (e.g. 60Zn and 100Sn) and heavy nuclides (e.g. 150Yb and210Th), the detection efficiencies increase much more slowly than those of thelight nuclides. For α particles and protons with energies larger than 1 MeV,both detection efficiencies can be considered to be 100%. For the implantationmethod where both the energies of the emitted particles and a part of theheavy recoil are detected, one needs to consider properly the energy loss of theheavy recoil in the detector. Some experimentalists have already noticed thiseffect and made the correction for their results [BJP+91, BAB+96, HHM+12].However, the partial deposited energy of the heavy recoil is not always con-sidered by others.

In the following, a concept about how to treat the calibration line andmake a correction to the published experimental results will be illustrated, incase the partial recoiling effect was not taken into account.

Here we take α decay as an example. If we consider the recoiling energy,the new scale should be adjusted to:

Ed = Eα + ER ×K, (4.14)

where Ed is the total detected energy, Eα is the kinetic energy of the α particle,ER is the recoiling energy andK is the detection efficiency for the recoil nuclideat energy ER. It is Ed that should be used in the energy calibration ratherthan Eα. The recoiling energy can be expressed as:

ER =4

M − 4Eα, (4.15)

where M is the mass number of the mother nuclide. Combining Eq. 4.14 andEq. 4.15, the pure α-particle energy can be obtained:

Eα =Ed

1 + 4KM−4

(4.16)

For proton-decay experiments where Qp is often used in the calibration(as one considers erroneously that the energies of the proton and of the heavyrecoil nuclide are fully detected at the same time), one can obtain a similarrelation as Eq. 4.14:

Ed = Ep + ER ×K (4.17)

where Ep is the proton energy and has the expression:

Ep =M − 1

MQp (4.18)

59


with recoiling energy in proton decay:

ER =1

MQp. (4.19)

Combining Eq. 4.17, 4.18 and 4.19, one can obtain:

Qp =M

M − 1 +KEd (4.20)

Applications

Here I will illustrate how to make corrections for the decay data, whenthe partial recoiling energy was not considered in experiments.

The case of 255Lrm(α)

In [HLMY+08], the detector was calibrated using a well-known α-particleenergy of 7923(4) keV in 216Th [LM16b]. The recoiling energy of the daughternuclide 212Ra is calculated as 7923 × 4/212 ≈ 150 keV and at this energythe detection efficiency K is 29.12%. The calibration line of 216Th shouldbe adjusted to Ed(216Th) = 7923 + 150 × 0.2912 = 7967 keV. In the α-decayspectrum, the α-particle energy of 255Lrm is 8371 keV, from which the detectedenergy of 255Lrm can be deduced as Ed(255Lr) = 7967×8371/7923 = 8417 keV.The recoiling energy of the α-decay daughter nuclide 251Md can be calculatedapproximately as 8417 ∗ 4/255 ≈ 131 keV and at this energy, its detectionefficiency is 29.08%. According to Eq. 4.16, the pure α-particle energy of 255Lrmis calculated to be 8378 keV. The difference between the published value andthe corrected value is 7(10) keV. The same routine can be applied to theα-decay energy of the 255Lr ground state.

The case of 69Kr(β+p)

In [SMB+14], the β-delayed proton-decay (β+p) energy of 69Kr was de-termined to be 2939(22) keV using known β-delayed proton decay energies of806, 1679, and 2692 keV for 20Mg and 1320, 2400, 2830, 3020, and 3650 keVfor 23Si. The authors assumed (erroneously) that the recoil energy would befully recorded at the same time [Mei15]. As one can see from Fig. 4.2 the de-tection efficiency for the intermediate nuclide, e.g. 60Zn, is between 30%∼40%and its neighbouring nuclides show similar behaviour. The recoiling energy ofthe β-delayed proton-decay 23Si at 3020 keV is 3020/23 ≈ 131 keV and thedetection efficiency for the decay daughter nuclide 22Mg is 59.75%. The effec-tively detected energy of this calibration line is 2967 keV according to Eq. 4.17.The detected energy of β-delayed proton-decay nuclide 69Kr is deduced to be2967 × 2939/3020 ≈ 2887 keV. The detection efficiency of the daughter nu-clide 68Se is 30.79% at the corresponding recoiling energy. Applying Eq. 4.20,

60


the β-delayed proton decay energy of 69Kr is calculated to be 2916 keV. Thedifference between the corrected value and the published one is 23(22) keV,which exceeds 1σ.

The case of 53Com(p)

In [SLS+15], the DSSD was calibrated using 41Ti β-delayed proton en-ergies Ep of 986(2), 1542(2), 3083(4), and 4735(3) keV from [CS01]. For theproton energy of Ep = 1542 keV from 41Sc, the energy of the recoiling nuclideis 1542/38 = 39 keV and the detection efficiency at this energy is 34.87%. Theeffectively detected energy is therefore 1542+39×0.3487 = 1555.5 keV. In thepublication the proton-decay energy Qp = 1558(8) keV of 53Com was obtainedfrom an observed line lying 23 keV lower than the 41Sc(p) Ep = 1542 keVline (supposed then to have deposited a total energy of Qp = 1581 keV). Thedeposited energy for the proton-decay of 53Com is Ed = 1558/1581× 1555.5 =1532.9 keV. The recoiling energy of 52Fe is approximatively 1558/52 = 30 keVand the detection effciency K at this energy is 31.76%. Thus, the proton-energy of 53Com was effectively Ep = 1532.9/(1 + 0.318/32) = 1523.7 keV andQp = 1553.0 keV, with uncertainty of 8 keV. We (M. Wang, G. Audi, andW. Huang) discussed this case with Shen et al. [SLS+15], and they agreedwith us on the general idea of the detection efficiency. They have re-calibratedthe detectors with the detection efficiencies and obtained the new value Qp of53Com, which is 1553.3 keV. Their result is in agreement with our correctedvalue, thus confirming the validity of our treatment.

From the three examples discussed above, we demonstrated that the re-coiling effect should not be ignored. In [HHM+12], the detection efficiency Kwas assumed to be 28% and was applied to all the calibration lines and thenuclide of interest. It is reasonable to use K = 28% universally in this case asone can see from Fig. 4.2 that K becomes almost constant for heavy nuclides.For light nuclides, where K differs quite a lot (59.75% for 22Mg and 30.79%for 68Se), one should treat them differently.

The correction method for the recoiling energy has been published in[HA17].

4.3 High precision α-decay energies

From the direct α-particle energy Eα measurement, one could obtain theQα value via Eq. 4.12. One should however recall that Eq. 4.12 was derivedfrom non-relativistic conditions and without consideration of atomic effects.Eq. 4.12 is valid only in α-spectroscopy measurements where a moderatelyaccurate energy scale is needed.

61

4.4. CORRELATION

The highest precision of α-decay energies comes from magnetic spec-trograph [GR71, RWK86, RW84] and they have been used for several rea-sons [RGG72]: a) they provide calibration points for all α spectra observedwith high resolving power; b) precise Q-value determinations of nuclear reac-tions are often based on α-energy standards; c) atomic mass difference maybe calculated accurately based on α energies.

In α decay, where the final atom is left in such a state but the emerging αparticle is a bare nucleus, the relation between the α-particle energy Eα andits α-decay energy Qα, taking relativistic and atomic effects into account, canbe written as:

Qα = (M −mα)±√

(M −mα)2 − 2 ·M · Eα +BHe, (4.21)

where M is the mass of the decaying parent nuclide, mα is the mass of αparticle and BHe is the two-electron binding energy in helium which is 79 eV.The derivation of Eq. 4.21 can be found in Appendix A.

When the decay takes place between the ground state of the parent andexcited state of the daughter nuclide, the decay-Q value should be revised to:

Qα = Q∗α − Ex (4.22)

where Q∗α represents the decay-Q value to an excited state Ex in the daughternuclide.

Table 4.2 lists some of the most precise α-particle energy data and theirdeduced values of Qα, using the classical formula and the relativistic one (plusatomic binding energy). It shows that the relativistic and the atomic correc-tions are indispensable: the difference between the two formulae, which is listedin the last column, is larger than 3σ.

Table 4.2: Accurate α-decay calculations. Col. 1 is the decay incident , Col. 2is the α energy, Col. 3 and Col. 4 are the calculated decay energies usingclassical (Eq. 4.12) and relativistic (Eq. 4.21) formulae, respectively, and col. 5is difference between the values from two formulae.

Item α energy (keV) Qα(cla) (keV) Qα(rel)(3) (keV) Diff (keV)148Gd(α)144Sm 3182.68(0.03) 3271.12(0.03) 3271.24(0.03) 0.12(0.03)252Cf(α)248Cm 6118.10(0.04) 6216.82(0.04) 6216.98(0.03) 0.16(0.04)253Es(α)249Bk 6632.51(0.05) 6739.10(0.05) 6739.28(0.05) 0.18(0.05)

4.4 CorrelationIt is a basic assumption, as mentioned in Section 3.1, that all the input

data is independent. Such a treatment is reasonable since correlations affectonly very few masses with high precision. However, even though the input data

62


are considered independent, the parameters, here the masses, which need to befitted, are correlated. This can be seen directly from the non-zero off-diagonalelements in the error matrix ∗ A−1.

The atomic masses have long been used as input data in the Committeeon Data for Science and Technology (CODATA) adjustments [MNT16b]. Thecorrelations among the input data, here the masses, will be considered by theCODATA evaluators. The discussion in this section was initiated by B. N.Taylor [Tay14].

Correlation coefficients between 28Si, 29Si and 30Si

In this section, for simplicity, the atomic mass of a nuclide X is denotedby A(X), the uncertainty is denoted by u(X), and the covariance between twonuclides X and Y is denoted by u(X, Y ). The correlation coefficient betweenX and Y can be expressed as:

r(X, Y ) =u(X, Y )

u(X)u(Y ). (4.23)

Note that the covariance of a mass A(X) with itself is the variance of A(X),which is the square of its uncertainty u(X). Also note that, in general, u(X, Y ) =u(Y,X).

The calculations here concern three nuclides 28Si, 29Si, and 30Si, whichare labeled r(28, 29), r(28, 30) and r(29, 30), respectively. From the covariancematrix, one reads u(28, 29) = 2.84 × 10−19. Combining u(28) = 5.24 × 10−10

and u(29) = 6.00 × 10−10, one obtains immediately the correlation between28Si and 29Si:

r(28, 29) =u(28, 29)

u(28)u(29)=

2.84× 10−19

5.24× 10−10 × 6.00× 10−10= 0.9033

However, the covariance matrix does not give the covariances involvingA(30), because A(30) is a secondary nuclide and was not included in theAME2016 least-squares adjustment.

To calculate the covariance u(29, 30), one notes that the only commoncomponent of uncertainty to A(29) and A(30) is the uncertainty of A(29)itself †. Thus, their covariance is just the product u(29) × u(29) or u2(29).Hence we have:

u(29, 30) = u(29, 29) = 3.60× 10−19,

which leads to a correlation coefficient of:

r(29, 30) =u(29, 29)

u(29)u(30)=u(29)

u(30)= 0.0803.

∗. The error matrix can be accessed easily though AMDC website"http://amdc.in2p3.fr/masstables/Ame2016/filel.html" under the file name "a0p5gqfu.zip"for the all masses involved in the least-squares procedure in 2016.†. The mass of 30Si is determined by 29Si through a (n,γ) reaction

63

http://amdc.in2p3.fr/masstables/Ame2016/filel.html

4.4. CORRELATION

In the calculation, we use u(30) = 7.47× 10−9.To calculate the covariance u(28, 30), we note that the source of the co-

variance of A(28) and A(29) is present in A(30) because A(30) is deriveddirectly from A(29) and there are no other common sources of uncertainty inA(28) and A(30) (otherwise A(30) would become primary). Hence we have:

u(28, 30) = u(28, 29) = 2.84× 10−19,

from which the correlation coefficient can be obtained:

r(28, 30) =u(28, 29)

u(28)u(30)= 0.073.

Correlations between adjusted masses are important for the CODATAgroup, because masses are entered directly as a part of input data in theiradjustment for the fundamental constants. A covariance matrix which containsfourteen nuclides were provided to the CODATA group after the publicationof AME2016.

64

Chapter 5

Mass Models

5.1 Semi-empirical approaches

An ideal theory of mass would be the one that reproduces the nuclearbinding energies from “real” nucleonic interactions, meaning solving the Schrödingerequation [Pea01]:

HΨ = EΨ, (5.1)

where

H = − h

2M

∑i

∇2i +

∑ij

Vij +∑ijk

Vijk. (5.2)

Here Vij and Vijk are two-body and three-body interactions, respectively. Thetotal energy of a nucleus E is equivalent to the binding energy B but withopposite sign (E = −B). The ab initio methods (ab initio is a Latin termmeaning “from the beginning”) aim at finding the solution of the Schrödingerequation for the atomic nucleus from nucleon-nucleon interactions. So far,calculations can only reach the tin region [MSS+18]. At the same time, thecalculation is not simple, since one has to deal with the strong short-range re-pulsion force between nucleons and the tensor coupling. Due to the difficultiesin the calculation, one has to resort to a phenomenological method, e.g., basedon phenomenological interactions, which can calculate the properties (here themass) for a vast number of nuclei, especially for those that are related to ther-process nucleosynthesis [Arn96].

Since we can not obtain the exact wave function Ψ in Eq. 5.1 for such amany-body system (nucleus), we have to refer to the semi-empirical approches:one can simplify the many-body problem, e.g., by replacing Ψ by a Slaterdeterminant Φ = det{φi(xi)} by properly antisymmetrizing the single parti-cle wave functions φi(xi) (Hartree Fock) using effective interactions, such asSkyrme forces [VB72] and Gogny forces [DG80]; or refine the Bethe-Weizsäckerformula by including shell corrections. The former method is microscopic andthe later one is macroscopic-microscopic.

65

5.2. ACCURACY

For the moment, the stellar nucleosynthesis models, in particular r-process,requires masses which cannot be produced by current facilities and one has torely on mass models. The choice of a mass model is important since a slightchange in mass (strictly speaking the mass difference between two adjacentnuclides) has a strong impact on energy generation and hence on astrophys-ical scenarios [Sch13]. Before using a model, its predictive power should beestimated carefully. The intention of this chapter is not to describe in detailevery component in each mass model. It only serves as an illustration of theability of the models in the description of experimental masses and their pre-dictive power. A detailed discussion of different models and applications canbe found in a review paper [LPT03].

Table 5.1: Information of eight mass models: the year of publication, thenumber of parameters in the model, and the mass table that was used tofit parameters.

Mass model Year of publication Number of parameters Mass tableETFSI-2 2000 9 AME1993FRDM95 1995 38 AME1993FRDM12 2012 41 AME2003KTUY05 2003 115? AME2003HFB26 2013 30 AME2012HFB27 2013 27 AME2012

WS4+RBF 2014 18+ AME2012DUZU 1995 33 AME1993

? Based on the original formulation from [KUTY00].+ To construct the smooth function S(X) with the RBF approach, the massesof 2148 experimentally known nuclides are considered. More details can befound in [WL11].

5.2 Accuracy

In this section, eight mass models are examined: Extended Thomas-Fermiplus Strutinsky Integral method (ETFSI-2) [Gor00], Finite Range DropletModel (FRDM95) [MNMS95] and its updated version with improved treat-ment of deformation (FRDM12) [MSIS16], a recent Weizsäcker-Skyrme plusRadial Basic Function (WS4+RBF) model [WLWM14], two recent Hartree-Fock-Bogoliubov mass models HFB26 [GCP13a] and HFB27 [GCP13b], theDuflo and Zuker (DUZU) model [DZ95], and the KTUY05 [KTUY05] model.Four models are macroscopic-microscopic (ETFSI-2, FRDM1995, FRDM2012,WS4+RBF), two are microscopic (HFB26, HFB27), and two are phenomeno-logical (DUZU, KTUY05). All mass models discussed here are semi empiricalin that some (or all) parameters are fitted to experimental masses. Table 5.1

66

CHAPTER 5. MASS MODELS

Figure 5.1: Masses of different Sn isotopes calculated from different modelswith respect to the mass model DUZU.

details the information related to each mass model.

A good mass model should on one hand reproduce the binding energies asseen in Fig. 1.1, and on the other hand have a good predictive power (see nextsection) when it extrapolates towards the unknown region. Fig. 5.1 representsthe deviations of different mass models with respect to the DUZU model forthe masses of Sn isotopes. Within the known region (between blue-dash lines),all models give similar results, since all models were fitted to the experimentalmasses. But when they go further away from the known region, they diverge:especially the neutron drip-line, different mass models predict different trends.Two mass models HFB26 and KTUY05 follow almost the same trend andfour mass models ETFSI-2, FRDM95, FRDM12 and HFB27 seem to grouptogether. The mass model WS4+RBF goes in the opposite direction withrespect to all the other mass models. We should notice that all models werepublished before AME2016, giving a good opportunity to check their predictivepower.

We will first study the model accuracy. The accuracy is the degree ofagreement between the masses calculated by a mass model and those fromexperiments. In order to study the model accuracy, we can define a root-mean-square deviation (rms) for a mass model:

67

5.2. ACCURACY

δrms =

√∑Nnuc

i=1 (mical −mi

exp)2

Nnuc

, (5.3)

where mical and mi

exp are the masses from a model and from an experiment,respectively, with the number of nuclides Nnuc included in the calculation. Therms deviation is an indicator of the accuracy for a mass model, and will beused in the following discussion ∗. Other quantities such a mean error δ whichis defined:

δ =

∑Nnuc

i=1 (mical −mi

exp)

Nnuc

, (5.4)

and the maximum deviation δmax, which calculates the largest deviation fromexperimental masses, is also used.

Table 5.2 lists the rms deviation δrms, mean deviation δ, and the maximumdeviation δmax for eight mass models for all nuclides with Z,N ≥ 8 with respectto three mass tables AME2003, AME2012, and AME2016. It is intriguing toinclude AME2003 and AME2012 for comparison: some models are relativelyold (dating back in 1995 as seen in Table 5.1) and their intrinsic robustnesscan be checked with time. To have a better vision of the accuracy for various

Table 5.2: Root-mean-square deviation δrms, mean deviation δ, and maximumdeviation δmax for nuclides with Z,N ≥ 8 with respect to three mass tablesAME2003, AME2012, and AME2016. The results for the eight mass modelsare listed.

AME2003Models FRDM95 FRDM12 KTUY05 ETFSI-2 HFB26 HFB27 WS4+RBF DUZUcount 2149 2149 2149 2048 2149 2148 2149 2149δrms 0.656 0.581 0.653 0.695 0.571 0.533 0.206 0.360δ −0.058 −0.004 −0.018 0.085 0.022 −0.021 −0.002 −0.009

δmax −3.783 −3.012 −3.052 −3.236 −2.698 −2.560 −1.441 −2.706AME2012

count 2353 2353 2353 2249 2353 2352 2353 2353δrms 0.654 0.579 0.701 0.679 0.564 0.512 0.170 0.394δ −0.059 −0.010 −0.058 −0.086 −0.006 0.005 0.000 −0.032

δmax 3.640 3.263 −2.932 −2.150 −2.715 −2.564 0.897 −3.060AME2016

count 2408 2408 2408 2300 2408 2407 2408 2408δrms 0.677 0.599 0.723 0.676 0.580 0.517 0.187 0.422δ −0.060 −0.009 −0.068 −0.087 −0.011 0.000 0.001 −0.032

δmax 3.998 4.568 −2.930 −2.150 −3.540 −2.554 1.485 3.458

mass models, the rms deviations are also plotted in Fig. 5.2. One can see from

∗. P. Möller introduced a “model error” [MN88]. But since it is not commonly used inthe literature, we omit it here.

68


Fig. 5.2 that the rms deviation is very similar for each mass model with respectto three mass tables. The mass model ETFSI-2 gives a δrms value of around0.7 MeV and DUZU gives a δrms around 0.4 MeV, which increases slowly withsuccessive mass tables. FRDM95, which was proposed before the publicationof AME2003, gives a δrms value of ∼ 0.7 MeV with respect to three mass tables.FRDM12, which used the same droplet model but with an improved treatmentof deformation and fewer approximations, gives a δrms value of around 0.6 MeV.The two microscopic mass models HFB27 (with the standard 10-parameterSkyrme force) and HFB26 (with two extra unconventional terms t4 and t5in the Skyrme force) give rms deviations lower than 0.6 MeV. WS4+RBFgives the lowest-ever rms deviation, which is below 0.2 MeV. KTUY05 gives aδrms of 650 keV compared to AME2003 and a slightly larger value of 720 keVcompared to AME2016. The very similar rms deviations for each mass modelcompared to different mass tables are due to the fact that, in AME2016, only17 masses have changed more than 500 keV compared to AME2003.

The average error δ is between several keV and tens of keV for all massmodels. The maximum deviation δmax is for most of the mass models largerthan 2 MeV, except for WS4+RBF, which reaches only 1.4 MeV.

Figure 5.2: Root-mean-square deviations of eight mass models for nuclidesZ,N ≥ 8 with respect to mass tables AME2003, AME2012, and AME2016.The number in the parenthesis indicates the number of parameters in thecorresponding mass model. Same illustrations will be applied to the successivefigures.

The global (Z,N ≥ 8) rms deviation displayed in Fig. 5.2 reflects theaverage accuracy of a model for the whole nuclear chart. It would howeverinduce a bias: a model with a good accuracy does not necessarily have the same

69

5.2. ACCURACY

quality for a specific region. It is demonstrated in [SLP18] that the accuracy ofa mass model strongly depends on the region of nuclei to which it is applied.Following the same routine as in [SLP18], the global region is divided into foursubregions: Light (8 ≤ Z < 28, N ≥ 8), Medium-I (28 ≤ Z < 50), Medium-II(50 ≤ Z < 82), and Heavy (Z ≥ 82). The calculations of δrms, δ and δrms

with respect to the experimentally known masses in AME2016 are listed inTable 5.3. Also listed are the number of nuclides in different regions.

Table 5.3: Root-mean-square deviation δrms, mean deviation δ, and maximumdeviation δrms from AME2016 in four regions: Light (8 ≤ Z < 28, N ≥ 8),Medium-I (28 ≤ Z < 50), Medium-II (50 ≤ Z < 82), and Heavy (Z ≥ 82).

LightModels FRDM95 FRDM12 KTUY05 ETFSI-2 HFB26 HFB27 WS4+RBF DUZUcount 350 350 350 242 350 350 350 350δrms 1.196 1.119 0.733 0.939 0.967 0.820 0.302 0.637δ −0.122 −0.017 −0.013 −0.603 0.033 −0.089 0.022 0.104

δmax 4.000 4.568 2.608 −2.150 −3.540 −2.554 1.485 3.458Medium-Icount 591 591 591 591 591 591 591 591δrms 0.687 0.620 0.824 0.582 0.520 0.579 0.194 0.428δ 0.051 −0.007 −0.346 0.002 0.030 0.143 −0.006 −0.019

δmax 2.441 2.301 −2.930 −1.752 1.648 2.016 1.105 −1.811Medium-IIcount 977 977 977 977 977 977 976 977δrms 0.478 0.370 0.550 0.641 0.455 0.390 0.148 0.330δ −0.136 −0.045 0.208 −0.065 −0.048 −0.045 0.003 −0.031

δmax −1.496 −1.330 −1.760 2.004 −1.730 1.523 0.494 −1.386Heavycount 490 490 490 490 490 490 490 490δrms 0.451 0.367 0.874 0.693 0.495 0.353 0.133 0.379δ 0.003 0.064 −0.209 0.017 −0.020 −0.017 −0.010 −0.026

δmax −1.999 −1.889 −2.587 −1.788 −1.764 −1.564 −0.412 −3.058

From Table 5.3 one can see that the rms deviation changes from regionto region. In the light region, δrms ranges from the lowest value of 0.302 MeV(WS4+RBF) to the highest one of 1.196 MeV (FRDM95). For a specific model,a strong variation of δrms can also be identified in different regions: HFB27 givesits lowest δrms of 0.353 MeV in the heavy region while the δrms increases to 0.816MeV in the light region. Fig. 5.3 displays the dependence of δrms on differentregions of eight models. As seen from Fig. 5.3, the rms deviations for the massmodels FRDM95, FRDM12, HFB26, HFB27, WS4+RBF, and DUZU have atendency to decrease towards the heavy mass region. A “saturation” of δrms isalso spotted in these models: their δrms remains almost unchanged when theypass from Medium-II to Heavy. However, a totally different trend is noted intwo mass models KTUY05 and ETFSI-2. KTUY05 presents the largest rmsdeviation of 874 keV in the heavy region, and its smallest δrms value of 550 keV

70


Figure 5.3: Root-mean-square deviations of eight mass models in four regions:Light (8 ≤ Z < 28, N ≥ 8), Medium-I (28 ≤ Z < 50) Medium-II (50 ≤ Z <82) and Heavy (Z ≥ 82).

in the meadium-II region. The minimum δrms value for ETFSI-2 is found inthe medium-I region, which is 582 keV.

The rms deviations are also displayed separately in different regions for allmodels in Fig. 5.4 (represented by green lines), together with the lines in blackdisplaying the Global accuracy . In the light region, FRDM95 gives the largestδrms value of 1.196 MeV and WS4+RBF gives the smallest value of 0.302 MeV.But all mass models have larger δrms than their global (Z,N ≥ 8) values. Itis probably due to the fact that the mean field on which all models dependis not sufficient to describe the light nuclides. When going to the medium-Iregion, δrms becomes smaller for all models, except for KTUY05, whose rmsdeviation increases to 0.824 MeV compared to 0.733 MeV in the light region.And in this region, the δrms overlaps with the Global δrms. Going further intothe medium-II region, the largest rms deviation is found for ETFSI-2 and thesmallest one for WS4+RBF. And the δrms values are smaller than the Globalδrms for all models. In the heavy region, the largest rms deviation is found forKTUY05 and the smallest one for WS4+RBF.

5.3 Predictive powerThe predictive power is the ability of a mass model to predict the unknown

masses. To some extent, the predictive power could be more important than the

71

5.3. PREDICTIVE POWER

(a) (b)

(c) (d)

Figure 5.4: Root-mean-square deviations from AME2016 in four regions (rep-resented by lines in green): (a) Light, (b) Medium-I, (c) Medium-II, and (d)Heavy. The Global rms deviation is also displayed (represented by black lines).

72


accuracy, since it can provide masses that cannot be produced at the currentfacilities, but are crucial in astrophysical calculations. To examine the relationbetween the accuracy and the predictive power, we can calculate the rmsdeviation separately: one is based on the experimental masses in AME2012(δrms(2012)), the other is based on the new measured masses in AME2016(δrms(new)) which were unknown in AME2012.

Table 5.4: Root-mean-square deviations in the Global (Z,N ≥ 8), Light (8 ≤Z < 28, N ≥ 8), Medium-I (28 ≤ Z < 50) Medium-II (50 ≤ Z < 82) andHeavy (Z ≥ 82) regions. The rms deviations are calculated separately forthe masses that were known in AME2012 δrms(2012) and for the new ones inAME2016 δrms(new).

GlobalModels FRDM95 FRDM12 KTUY05 ETFSI-2 HFB26 HFB27 WS4+RBF DUZUcount(2012) 2353 2353 2353 2249 2353 2352 2353 2353count(new) 61 61 61 57 61 61 61 61δrms(2012) 0.654 0.579 0.701 0.679 0.564 0.512 0.170 0.394δrms(new) 1.231 1.096 1.279 0.731 0.986 0.779 0.489 1.009

Lightcount(2012) 335 335 335 216 335 335 335 335count(new) 17 17 17 13 17 17 17 17δrms(2012) 1.144 1.056 0.692 0.959 0.926 0.791 0.247 0.546δrms(new) 1.851 1.879 1.310 0.700 1.489 1.157 0.790 1.528

Medium-Icount(2012) 575 575 575 575 575 575 575 575count(new) 18 18 18 18 18 18 18 18δrms(2012) 0.664 0.618 0.783 0.583 0.516 0.593 0.175 0.406δrms(new) 1.210 0.729 1.571 0.859 0.662 0.726 0.430 0.892

Medium-IIcount(2012) 970 970 970 970 970 969 970 970count(new) 7 7 7 7 7 7 7 7δrms(2012) 0.475 0.368 0.542 0.643 0.449 0.389 0.148 0.328δrms(new) 0.660 0.513 1.054 0.469 0.904 0.525 0.194 0.678

Heavycount(2012) 473 473 473 473 473 473 473 473count(new) 19 19 19 19 19 19 19 19δrms(2012) 0.448 0.367 0.870 0.694 0.489 0.352 0.133 0.376δrms(new) 0.530 0.360 0.992 0.703 0.655 0.403 0.161 0.525

The calculation of δrms based on two different sets is listed in Table 5.4.The dependence of δrms(2012) and δrms(new) for different mass models in dif-ferent regions are illustrated in Fig. 5.5. One can see from Fig. 5.5a that inthe global region, the condition of good predictive power defined by [SLP18]:

rms(new) ≈ rms(2012), (5.5)

is only roughly achieved by ETFSI-2 (but with a large value of ∼ 700 keV),while other models present a much larger δrms(new) value than that of theirδrms(2012). For example, FRDM12 gives a δrms(2012) value of 0.579 MeV but

73


gives 1.096 MeV for δrms(new). In the light region, ETFSI-2 shows the bestpredictive power (0.7 MeV) compared to other mass models. In the medium-I region, the predictive power of the mass models is becoming better andthe difference between δrms(2012) and δrms(new) is becoming smaller, exceptfor KTUY05. In the medium-II region, the difference between δrms(2012) andδrms(new) is strongly compressed for all models except for KTUY05, DUZUand HFB26. In the heavy region, two curves are almost overlapped, meaningthat the predictive power of all the mass models are compatible with theiraccuracy. One can refer to [SLP18] for the discussions of the predictive powerof other models.

In this chapter, the accuracy and predictive power of eight mass modelsare studied. The mass models, regardless of their intrinsic characters, couldreproduce the experimentally known masses but predict different behaviorwhen they extrapolate towards unknown regions.

The rms deviations for all the mass models are well below 0.8 MeV com-pared to three mass tables AME2003, AME2012, and AME2016. To checkthe accuracy in different mass regions, we divide the global region into foursubregions. We observe that going from the light region to the heavy region,the rms deviation for most of the mass models is becoming smaller, meaningthat their accuracy is becoming better. But it is not the case for the massmodel KTUY05, which presents the largest rms deviation in the heavy region.To estimate the predictive power, we use two sets of data for comparison,the mass table AME2012 and the newly measured masses in AME2016. Wefind that the predictive power of mass models depends strongly on the con-sidered regions. In the global region, the condition rms(new)≈rms(2012) isonly roughly fulfilled by the mass model ETFSI-2, while other mass modelspresent a much large rms(new) value than that of their rms(2012). We alsoobserve that ETFSI-2 has the best predictive power in the light region, whilein other regions, the mass model WS4+RBF has the best predictive power. Inthe heavy region, all the mass models find their lowest rms(new) value, andthe predictive power is compatible with the accuracy.

In general, phenomenological models could well reproduce known masses,but extrapolate badly; microscopic models have slightly larger rms deviationsbut show better extrapolation ability. The more fundamental basis that amodel relies on, the better chance it would have in reproducing actual masses.Fig. 5.6 displays the deviations of calculated masses and the experimentalones in AME2016. With Fig. 5.6, we can examine the discrepancies in sub-tle regions. For example, FRDM95, FRDM12, and DUZU have worse pre-dictive power (δrms > 2 MeV) in the neutron-rich calcium (Z = 20) region,while ETFSI-2 presents δrms ≈ 0.5 MeV. The shell and deformation effects arealso recognized. For example, in the A ∼ 100 (Z ∼ 38) deformation region,FRDM95, FRDM12, ETFSI-2, HFB26, and HFB27 over-predict the masses.

To qualify a mass model, the rms deviation should not be the only concern.Other nuclear ingredients such as Q-values, half-lives, spin and parity, fission

74


barriers, ect., should be also provided by mass models. The application of amodern mass model can also be found in astrophysics, such as the elucidationof the r-process nucleosynthesis [GCP16].

Next, a more reliable way to estimate the masses in Ame, by observingthe smoothness of mass surface, will be discussed.

75


(a) (b)

(c) (d)

(e)

Figure 5.5: δrms(2012) and δrms(new) in (a) Global (b) Light (c) Medium-I (d)Medium-II and (e) Heavy regions.

76


(a) Display of the deviations between the calculated masses and experimental ones in AME2016in a color plot for the mass model FRDM95.

(b) Same as Fig. 5.6a but for the mass model FRDM12.77


(c) Same as Fig. 5.6a but for the mass model KTUY05.

(d) Same as Fig. 5.6a but for the mass model ETFSI-2.78


(e) Same as Fig. 5.6a but for the mass model HFB26.

(f) Same as Fig. 5.6a but for the mass model HFB27. 79


(g) Same as Fig. 5.6a but for the mass model WS4+RBF.

(h) Same as Fig. 5.6a but for the mass model DUZU.

Figure 5.6: Display of deviations between masses from models and that fromAME2016 in color plots.

80

Chapter 6

Mass Extrapolation

6.1 Regularity of the Mass Surface

When atomic masses are displayed as a function of N and Z, we obtaina surface in a 3-dimensional space. However, the surface, which is subject toodd-even staggering effect caused by pairing, is rather uneven. If one dividesthe surface into four sheets (e-e, e-o, o-e and o-o, e-o means even in N and oddin Z), one would immediately note that the e-e sheet lies in the lowest, the o-olies in the highest and the e-o and o-e sheets lie nearly half way between the e-eand o-o sheets. The separation of four sheets, where each of which has a smoothbehavior, minimizes the pairing effects. Therefore, the smoothness (regularity)is observed to be a basic property of the mass surface and can help derivingunknown masses from measured ones. Thus, dependable short-range estimatesof unknown or poorly known masses can be obtained by extrapolating wellfrom well-known masses on the same sheet. However, the smoothness couldalso be interrupted by a sudden change of physical quantities (shell closure,deformation, etc.).

The extrapolation is used for several purposes:a) A coherent deviation from regularity indicates a change of physical

properties in a region. If only a single mass violates the trends from the masssurface (TMS) while its neighbouring nuclides show a regular character, itscorrectness would be questioned.

b) For two measurements using different methods which conflict with eachother and no conclusion can be drawn, the one that agrees with TMS wouldhelp decide which one to be accepted or rejected.

c) For some extremely exotic nuclides whose masses were measured onlyonce, their values should be checked against the estimate form the TMS.

d) We want to include a number of nuclides occurring in reactions anddecays with precise energy values but having no connection to any knownmass.

e) Drawings of the mass surface allows to derive estimates for still-unknown

81

6.2. SCRUTINIZING THE MASS SURFACE

masses, which serves as a guide for future experiments.In addition, the masses of all nuclides having the same iso-parameters of

N,Z,A = N + Z and N − Z lying between known nuclides would also beobtained in the same way.

6.2 Scrutinizing the Mass Surface

Direct representations of masses in a 3-dimensional space is not conve-nient, since the mass surface has a very large expansion along the mass axis(mass number ranging from 1 u to about 300 u). In the work of [WAH88], theextrapolation was done by scrutinizing the mass surface in four sheets. Wap-stra et al. [WAH88] analyzed the differences between the differences betweenexperimental masses and an expression obtained by a proper consideration ofpairing effects [JHJ84] with a suitable smooth function f(Z,N). Such a treat-ment allowed to examine the mass surface on almost the same scale (aroundseveral MeV) and remedy some oscillations caused by pairing. However, theprocedure was rather complicated (see Fig. 1 in [WAH88]). And the separatesheets might diverge, which makes such extrapolation unpredictable.

An alternative to bypass these difficulties is to look at the derivativesof the mass surface. By derivative, we mean a specific difference between themasses of two nearby nuclides. The derivatives preserve the smooth behaviorwhich extends from the known masses to the unknown ones on one hand, andmagnify the local structure on the other hand. The pioneering work of [Wap65]aimed at estimating unknown masses based on the studies of five derivatives :two-neutron and two-proton separation energies, α-decay energies, beta-decayand double-beta decay energies. The primary intention was to retain reactionand decay results with known energies but having no connection to the knownmasses. A suspect mass which ruins the derivative plots can also be spottedeasily.

An “Interactive Graphical” (I-G) tool was devised [BA93] in the 1990s forobserving different derivatives of mass surface. It can display four derivativessimultaneously in the same graph, which allows to study constraints super-imposed by different derivatives, i.e. the smooth property should be fulfilledat each quadrant at the same time. Fig. 6.1 is a screen shot from the (I-G)tool for four derivatives : two-proton separation energy S2n, two-neutron sep-aration energy S2p, α-decay energy Qα, and double-beta-decay energy Qββ.Each point represents a nuclide and is connected by iso-parameters Z, N , Zand Z, respectively. Vertical lines at each point represent the correspondinguncertainty. As mentioned before, the extrapolation is based on the assump-tion that the mass surface is smooth and it is indeed true: in the S2n quadrant(upper left in Fig. 6.1), their values decrease smoothly with N from N = 62to N = 72. The change of a physical property can also be noted: a suddenchange of the values of S2p (upper right in Fig. 6.1) at Z = 50 depicts a shell

82

CHAPTER 6. MASS EXTRAPOLATION

closure. Based on the regular smoothness of the mass surface, an experimentalmass which ruins the smootheness of the mass surface should be checked withcaution and, if necessary, should be replaced by an estimated value (with label#). In AME2016, 31 cases which violates the smootheness of the mass surfacehave been replaced by values from the trends from the mass surface (TMS)(see Table C in [HAW+17]).

Since we are dealing with derivatives which involve two nuclides, i.e.each point is the difference between two masses, a deviation from the reg-ular surface could be either due to the related mass itself or its connectedpartner. Other constraints, apart from the requirement that all derivativesshould be compatible with each other, should be also imposed. Two alge-braic constraints are mainly considered here. First, if the masses of two nu-clides (Z,N) and (Z,N + 4) are known but not the mass (Z,N + 2) thenthe choice of S2n(Z,N + 2) and S2n(Z,N + 4) is not random, since the sumof the two is known. Secondly, two α-decay energies Qα(A + 4i, Z + 2i) andQα(A − 2 + 4i, Z + 2i) (i is an integer number) may be known and thus thedifference between the two is the two-neutron separation energy of a nuclidewith (A+ 4i, Z + 2i).

Other derivatives such as one-proton and one-neutron separation energiescould also be used in the mass extrapolation. In this case, derivatives shouldbe plotted separately according to different parities.

6.3 Subtracting a mass model from the experi-mental mass surface

If we consider the difference between two nearby masses from a massmodel instead of absolute masses, one would immediately find that the differ-ence between any two models becomes much smaller. Fig. 6.2 illustrates thecalculation of two-neutron separation for Sn isotopic chains based on the massmodels discussed above. The calculation shows a similar trend for all massmodels, even when they extend to the unknown region. It is probably due tothe fact the different nature of models are to some extent smeared out whenone expresses mass difference of adjacent nuclides instead of absolute masses.Moreover, all models predict well the shell closures at N = 50 and N = 82,shown with black-dash lines in Fig. 6.2.

It was first noticed by Wapstra [BW72] that, for practical use, a massformula should not (actually cannot) reproduce absolute masses as well as pos-sible, but rather the mass differences (separation energies, α-decay energies,etc.). A better solution for introducing a mass formula in the mass extrapo-lation is to subtract the actual masses from a “good” mass model. The studyof these differences is another way to perform extrapolation. However, choos-ing such a model is not straightforward: a mass model should have a good

83

6.3. SUBTRACTING A MASS MODEL FROM THE EXPERIMENTAL MASSSURFACE

Figure 6.1: Screen shot of the “Interactive Graphical” tool displaying fourderivative of the mass surface: two-neutron separation energy S2n, two-protonseparation energy S2p, α-decay energy Qα and double-beta-decay energy Qββ

from upper left to bottom right. The lines between two points have the sameiso-properties Z,N , Z and Z, respectively. A universal smoothness is identifiedin each quadrant, except when it comes across a shell closure: in the S2p plotat Z = 50 (a sudden drop of slope).

84


Figure 6.2: Two-neutron separation energies of Sn isotopes for all models dis-cussed in Chapter 5. Two vertical dash lines signify magic numbers N = 50and N = 82.

treatment of all phenomena in a nucleus (deformation, Wigner effect, pairing,etc.) as complete as possible by adjusting its model parameters to the actualmasses, which gives an overall satisfactory prediction. Thirteen mass modelswere studied using the I-G tool [BA93] and the DUZU model with sphericalbasis (DZ10sph) [DZ96] was chosen to be such a preferred model, because themass surface can be displayed much more smoothly with this mass model.

The combination of derivatives and the use of difference between actualmasses and the masses from a model enables practical extrapolation. As men-tioned above, every point in the derivative plots involve two masses and onehas to find out which one is the “culprit” that is responsible for the derailedpoint in any derivative plot. If one works on the mass difference plot, one canmanipulate every single mass more easily.

The modification of any single point in one of the quadrant of the I-G plotwill automatically update the other plots. Fig. 6.3 displays the experimentalmasses subtracted by the spherical DUZU model (DZ10sph) as a function of Nand Z, the experimental two-neutron separation energies, and the experimen-tal two-proton separation energies in four quadrants. It is rather remarkable tosee that in the upper two quadrants the trends are very flat: the shell closurehas been well considered by DZ10sph and no sudden drop in the plot is seen.As DZ10sph used the same shell strength for all nuclides at N = 50, a con-tinuous increasing trend is noticed in the upper-left quadrant in Fig. 6.3 for

85


Figure 6.3: Display of the differences between experimental masses and theDUZU model as a function of N and Z, of the experimental two-neutronseparation energies, and of the experimental two-proton separation energies infour quadrants.

86


decreasing Z due to shell quenching. However, such trends, as long as regularand smooth, would instead help extrapolation.

The extrapolation, which is based on the observation of different deriva-tives and the difference between actual masses and a formula, has been provento be the most powerful tool to obtain still-unknown masses that are not too far(two or three mass unit) away from the last known mass. The root-mean-squaredeviation for all estimated masses in AME2012 that are known in AME2016is 0.396 MeV (55 cases in total), which is smaller than any of the the massmodel discussed in the previous chapter (see Table 5.4). However, such extrap-olation, which strongly depends on the knowledge of the last known masses,has obvious shortcoming: if the mass value of the last known nuclide is wrong,and based on which the extrapolation is performed, erroneous extrapolationwould propagate towards the unknown region.

For example, the masses of 77−79Cu were measured previously with pre-cisions of 500 keV [HBGU+06]. However, the TMS suggested that 77Cu and79Cu should be 320 keV and 1760 keV more bound, respectively. In AME2016,their results of 77Cu and 79Cu were replaced by estimated values (see Table Cin [HAW+17]). Only the result of 78Cu was used.

Recently, the masses of copper isotopes 75−79Cu have been remeasured bythe ISOLTRAP mass spectrometer [WAA+17]. Their resulting masses showthat 77Cu and 79Cu are more bound by 240 keV and 670 keV, respectively, com-pared with the estimated values in AME2016. The extrapolation in AME2016for the Copper isotopic chain was based on the mass values of 76Cu [GAB+07]and a poorly known mass of 78Cu [HBGU+06]. The results from [WAA+17]confirms the 76Cu mass in [GAB+07], while disagreeing with 78Cu in [HBGU+06]by ∼ 300 keV. For extrapolation, we first replace the poorly known mass(78Cu) and estimated masses (77Cu and 79Cu) by the new ISOLTRAP results[WAA+17]. And then extend the trend of the copper chain following that ofthe zinc isotopic chain. Since the masses of 80−82Zn were well known, we knownroughly the position of successive copper masses after crossing theN = 50 shellclosure. The new mass surface is illustrated in Fig. 6.4, where the blue curverepresents the new extrapolation. Based on the new results from [WAA+17],the estimated masses of 80Cu, 81Cu, and 80Cu have been reduced by 480 keV,490 keV, and 410 keV, respectively, compared with the AME2016 extrapo-lation. The isotopic chains of nickel, cobalt, iron, manganese, chromium andvanadium are also affected).

In this chapter, the extrapolation method in Ame is described. Based onthe smooth and regular observation of the mass surface, we can assign a massto an unknown nuclide, given the regular behavior extends to the unknownregions. Such extrapolation routine strongly depends on the knowledge of thelast known nuclide based on which the extrapolation is performed. In principle,Ame make an estimate for the nuclide which has been observed or provento exist. We notice that the predictive power of the Ame extrapolation wasstudied previously in the literature (see Table I in [LPT03]) and the authors

87


Figure 6.4: Four derivatives the same as Fig. 6.3 but zooming around 79Cu.New extrapolation (in blue) based on the results from [WAA+17].

88


demonstrated that the systematics-based predictions in Ame are particularlyaccurate.

89


90

Chapter 7

Experiments

7.1 Principle of Ion Traps

The advantages of trapping ions are threefold. First, this allows observingthe ion motion for an extended period of time until they decay. In this case,we can make the best use of the ions, especially for those that are producedin minute quantity. Secondly, the ions are confined in a small volume thus theinhomogeneity of the magnetic field has less effect on the stored-ion frequency.Thirdly, one can manipulate the stored ions using an external circuit for variouspurposes, such as cooling and excitation.

To achieve the confinement of ions in space, a potential minimum is re-quired in three dimensions. A desirable confining force is the one which linearlydepends on the distance between the stored ions and the center of the trap.This results in a harmonic potential for the confined ions. Fig. 7.1(b) displaysthe trapping configuration of a Penning trap, where a DC potential (providingaxial confinement) is applied between a ring electrode and two end electrodesof hyperbolic shape. The direction of the homogeneous magnetic field (pro-viding radial confinement) is aligned with the rotational symmetry. Ions canalso be trapped without magnetic field. In this case, a radio-frequency (RF)voltage is applied between the ring electrode and the end caps (Fig. 7.1 (a)).Such a trap is called Paul trap or radio-frequency quadrupole (RFQ) trap.Besides the hyperbolic geometry, cylindrical Penning traps (Fig. 7.1 (c)) canalso provide a quadrupole potential by applying appropriate voltages betweenring segments.

Ion Motion in a Penning trap

The ions in a magnetic field and a quadrupole electric potential performthree independent eigenmotions [BG86]: the harmonic oscillation along thetrap axis at the axial oscillation frequency ωz, the modified cyclotron motionat frequency ω+ and the magnetron motion at frequency ω−. The latter two

91

7.1. PRINCIPLE OF ION TRAPS

Figure 7.1: Hyperbolic electrode geometry of Paul trap (a) and Penning trap(b). Trapping of charged ions can be realized by applying a voltage differencebetween the ring electrode and the end electrodes. Penning traps can also havecylindrical electrodes (c). Figure from [Bla06].

motions are radial motions and perpendicular to the trap axis. Fig. 7.2 displaysthe ion motion in a Penning trap.

The three eigenfrequencies can be written as [BG86]:

ωz =

√qUdcmd2

, (7.1)

ω+ =ωc2

+

√ω2c

4− ω2

z

2, (7.2)

ω− =ωc2−√ω2c

4− ω2

z

2, (7.3)

where Udc is the voltage applied between the ring electrode and the two end

caps, and d is the dimension of the trap d =

√(z2

0) +ρ202/2 (2ρ0 and 2z0 are the

inner ring diameter and the closest distance between the end caps, respectively,see Fig. 7.1 (b)).

By comparing the three eigenfrequencies, one can obtain the relations be-tween the cyclotron frequency and the eigenfrequencies in a perfect quadrupolefield:

ωc = ω+ + ω−, (7.4)

andω2c = ω2

+ + ω2− + ω2

z . (7.5)

92

CHAPTER 7. EXPERIMENTS

Figure 7.2: Sketch of ion motion in a Penning trap.

Thus, the mass of an ion can be determined either by measuring the sumof the two radial frequencies or by measuring the three independent eigen-frequencies. In this thesis, the relation in Eq. 7.4 is used to determine themasses.

Cooling

Due to the imperfections of the real Penning trap, such as the openingholes for ion injection and ejection, or electrodes not extending infinitely, whichwould induce a higher-order multipole electric field, which renders Eq. 7.4 in-valid, cooling is essential. Various ion cooling techniques such as resistive cool-ing, buffer gas cooling, laser cooling, etc., can reduce the amplitude of the ionmotion and thus confine ions in a smaller volume. This is important for per-forming high-accuracy measurements, since ions probe less the imperfectionsof the electric and magnetic fields. For radioactive nuclides, buffer gas coolingis generally used. Since noble gases have high excitation potential, they areideal choice to cool ions. By collisions with noble gas, such as helium, thehot ions lose energy and their motion can be damped. However, the situationin Penning traps becomes more complex. During cooling, the amplitudes ofthe axial and modified cyclotron motions will decrease while the amplitude ofthe magnetron motion ∗ will increase. Finally, the ions will hit the ring elec-trode and be lost. To counteract the outward radial diffusion caused by thebuffer gas, one can use the side-band cooling technique [SBB+91]. By apply-ing an RF voltage to the segmented ring electrode at the cyclotron frequencyωc = ω+ + ω− , the radial motions will couple with each other, i.e. the con-version between the modified cyclotron motion and the magnetron motion

∗. The maximum energy of the magnetron motion is at the center of the trap.

93

7.1. PRINCIPLE OF ION TRAPS

will take place periodically [BMSS90]. Since the modified cyclotron motion isdamped faster than the magnetron motion, the ions can be recentered after acertain cooling time. Side-band cooling is mass selective and can be used torecenter the ions of interest and eliminate isobaric contaminations.

Time-of-flight detection of the cyclotron resonance

There are various types of techniques to determine the cyclotron fre-quency. The one that is widely used for the radioactive nuclides is the time-of-flight detection of the ejected ion cyclotron resonance (TOF-ICR), whichwas first applied to the measurement of the mass ratio between proton andelectron [GKT80]. The idea of TOF-ICR is to measure the cyclotron energy bymeasuring the time of flight of the ions through an inhomogeneous magneticfield. This technique, though destructive, is suitable for short-lived nuclides.

After being captured in the Penning trap, a dipole excitation is appliedto the magnetron motion of the stored ions. After a certain period of time,the ions are first driven to a magnetron radius ρ−. An azimuthal quadrupolerf-excitation is then applied for a time period Trf at the sum of the radialfrequencies ωc = ω+ +ω−, which converts the magnetron motion to the modi-fied cyclotron motion [KBK+95]. After full conversion, the initial magnetronradius ρ− is transferred completely to the modified cyclotron radius ρ+. Asthe frequency of the modified cyclotron motion is generally much higher thanthat of the magnetron motion, the radial energy of the ion can be expressed asEr(ωrf ) ≈ E+(ωrf ) = m/2ω2

+ρ2+(ωrf ). At resonance, the ions gain maximum

energy (maximum ρ+) while in the case of off-resonance, the ions gain lessenergy. After excitation, the ions are released axially from the center of thetrap to a MCP detector, which is placed 1.2 m above. In the Penning trap, themagnetic moment of the ions ~µ(ωrf ) = (Er/B)z is proportional to the radialenergy and is conserved after ejection if the magnetic field in the drift pathchanges slowly. When ions pass through the magnetic field gradient outsidethe Penning trap, the interaction between the magnetic moment and the mag-netic field gives rise to an accelerating force ~F = −~µ(ωrf )(~∇ ~B) on the ions.The detection technique is illustrated in Fig. 7.3

Therefore, if the rf field is at resonance with the ion cyclotron frequency,the ion will gain more energy and reach the detector faster than the off-resonance ions. This can be seen (see Fig. 7.4) from the minimum of thetime of flight as a function of the excitation frequency.

A theoretical description of the line shape of the TOF from the trap centerz0 to the detector position z1 for a given radial energy Er as a function of theapplied rf frequency is given in [KBK+95]:

T (ωrf ) =

∫ z1

z0

√m

2(E0 − q · V (z)− µ(ωrf ) ·B(z))dz, (7.6)

94


Figure 7.3: Illustration of the TOF-ICR technique

where E0 denotes the initial axial energy, Vz and Bz the electric and magneticpotential along the drift path, respectively.

7.2 Experimental setup at ISOLTRAP

The Penning trap mass spectrometer ISOLTRAP is located at the Isotopemass Separator On-Line facility ISOLDE at CERN. Radioactive nuclides areproduced by the bombardment of a thick, heated target with 1.4-GeV protonbeam from the CERN proton synchrotron (PS) booster. The resulting nuclidesthen pass through a transfer line and are ionized using different ionizationtechniques such as surface ionization, laser ionization (Resonance IonizationLaser Ion Source RILIS), or plasma ionization, depending on the chemicalproperties of the nuclides under investigation. After ionization, the ions areaccelerated and mass-separated either by the High Resolution Separator (HRS)or General Purpose Separator (GPS), with mass resolving power of 5000 and1000, respectively. The ion beam is then transported to the ISOLTRAP setup,see Fig 7.4

The ISOLTRAP setup consists of four traps [KAB+13]: a linear radio-frequency quadrupole cooler and buncher (RFQ), a multi-reflection time-of-flight mass separator/spectrometer (MR-TOF MS), a cylindrical preparationPenning trap and a hyperbolic precision Penning trap. The RFQ is placed ata voltage-flotable platform and used for stopping, cooling, and bunching thecontinuous ion beam with energy between 30 ∼ 50 keV from ISOLDE. Aftera few milliseconds of accumulation, an ion bunch is ejected from the RFQwith a typical kinetic energy of Etrans ≈ 3 keV and transferred to the MR-TOF MS. In MR-TOF MS, the ions can be trapped by using the in-trap lift

95

7.3. DATA ANALYSIS AND DISCUSSIONS

Figure 7.4: Schematic view of the ISOLTRAP setup [KAB+13]. See text fordetails.

technique [WMRS12], which reduces the kinetic energy of the ions to Etrap ≈2 keV. The contaminants can be separated after several hundred reflectionsbetween two electrostatic mirrors, based on the different mass-to-charge ratio.The ion beam is then ejected from the MR-TOF MS. If the ion beam isstrongly contaminated by isobars, one can use a Bradbury-Nielsen gate tofurther suppress the contaminants. Subsequently, the ion beam is transferredupwards to the preparation Penning trap. Here the isobaric contaminants areremoved by the mass-selective buffer gas cooling technique discussed above.Finally, the purified ion beam is transferred to the precision Penning trap forhigh-precision mass measurements.

7.3 Data analysis and discussions

The determination of the mass is carried out by measuring the cyclotronfrequency of the stored ions:

νc =1

2π· qBm. (7.7)

96


In the precision Penning trap, a dipole excitation is first applied to theion at its magnetron frequency, which brings the ions to a radius of about0.7 mm. And then a quadrupole rf excitation at νc = ν+ + ν− is applied for anexcitation time Trf to the ion, which converts the magnetron motion to thereduced cyclotron motion. The cyclotron frequency of the ions of interest isextracted by fitting the data by the theoretical line shape in Eq. 7.6. To obtainthe time-of-flight spectrum, the frequency of the rf field has to be scannedaround the ion’s cyclotron frequency. The time of flight of the ions from thecenter of the precision Penning trap to the MCP detector is recorded.

A TOF-ICR spectrum of 178Yb is displayed in the insert figure in Fig. 7.4,where the excitation time was set to Trf = 1.2 s. The x-axis denotes thequadrupole rf-excitation frequency and y-axis represents the mean time offlight. The black points represent the experimental data and the red curveis the theoretical line shape in Eq. 7.6. To calibrate the magnetic field, thecyclotron frequency νc,ref of an ion with well-known mass, either from analkali ion source or a laser ablation ion source, is measured before and afterthe measurement of the ion of interest. To compensate the slow drift of themagnetic field, the cyclotron frequency of the reference ion is interpolated tothe time when the ion of interest is measured.

Table 7.1: Experimental details in the production of the ions of interest. Listedare the experiment date, the target, the ionization technique, the ion energyfrom ISOLDE, and the mass separator used.

Species Date Target Ion source Energy Separator168Lu Jun 2011 Ta W surface 50 keV GPS178Yb Oct 2011 Ta RILIS 30 keV HRS

160Yb, 140CeO, 140NdO, 156Dy Aug 2014 Ta W surface 30 keV GPS52,55−57Cr, 55Mn, 59Fe Oct-Nov 2014 UCx Ta surface 30 keV HRS

75,77−79Ga Jun 2015 UCx-n Ta surface 30 keV HRS52Cr

Apr 2016 UCx

Ta surface

30 keV HRS53Cr RILIS54Cr

Ta surface55Mn

The atomic mass of the nuclide of interest m can be derived by:

m = r · (mref −me) +me, with r =νc,refνc

, (7.8)

wheremref is the mass of the reference nuclide andme is the electron mass. Theelectron binding energy can be neglected in the current analysis. Apart fromthe statistical uncertainty, the known error and uncertainty have to be takeninto account. The two main contributions are the mass-dependent shift and

97


the short-term fluctuation of the magnetic field [KBB+03]. The imperfectionof the electric quadrupole field and the misalignment of the precision trap axiswith respect to the the magnetic field axis can bring about a frequency shift,if the masses of the ion of interest and the reference ion are not the same. Themagnitude of the mass-dependent shift was determined to be [KBB+03]:

εm(r)

r= −1.6(4)× 10−10/u× (m−mref ). (7.9)

This correction should be applied to the frequency ratio r. In the currentanalysis, the absolute value of εm(r)

ris also added quadratically to the statistical

uncertainty of the frequency ratio r.Another source of uncertainties comes from the magnetic-field drift. Due

to the change of temperature, pressure, and the ferromagnetic materials nearthe magnet, the strength of the magnetic field can change over time. This effectcan be minimized by shortening the time interval between the measurements ofthe frequencies of the ions of interest and the reference ion. However, it wouldonly eliminate the long-term, slow decay of the magnetic field. The magnitudeof the short-term fluctuation was determined to be [KBB+03]:

uB(νref )

νref= 6.35(45)× 10−11/min×∆T, (7.10)

where ∆T is the time interval between the two frequency measurements of thereference ions. After considering all the known uncertainties, the systematicuncertainty was determined from the cross-reference carbon cluster measure-ments to be [KBB+03]:

uref (r)

r= 8× 10−9. (7.11)

This is the current precision that ISOLTRAP can reach for the TOF-ICRtechnique.

Besides the traditional TOF-ICR technique, Ramsey’s method of sepa-rated oscillatory fields [GBB+07] were employed to excite the ion’s cyclotronmotion. The idea of the Ramsey-type excitation is to use two rf pulses sepa-rated by a waiting time. Such excitation scheme allows reducing the line widthby around 60% and the statistical uncertainty of the cyclotron frequency by afactor of three, given the excitation time and the number of the recorded ionsare the same [GBB+07]. Fig 7.5a displays the TOF spectrum of 57Cr using theRamsey-type excitation, where the excitation time is 100 ms for both rf pulsesand the waiting time is 1 s. Prior to performing the Ramsey-type excitation,the cyclotron frequency should be known. In this case, a traditional TOF-ICRspectrum was taken (see Fig. 7.5b). As we can see from the Fig. 7.5a, the TOFspectrum is symmetric with respect to the central frequency. Moreover, theRamsey fringes are more prominent than the sidebands in Fig. 7.5b, whichpermits to determine the cyclotron frequency more precisely. The red curverepresents the theoretical line shape for the Ramsey scheme [Kre07].

98


The newly developed Phase-Imaging Ion-Cyclotron-Resonance technique[EBB+13] has been tested recently at ISOLTRAP [Kar17].

The mass measurements in this thesis stem from several experimentalcampaigns between 2011 and 2016. The related information is listed in Ta-ble 7.1. The cyclotron frequency ratios of the ions of interest and the relatedreference ion are listed in Table 7.2, together with the derived masses of the nu-clides. Since some of the data were already included in AME2016, values fromthe mass table of AME2012 are given instead for comparison. The half-livesof all the nuclides, extracted from [AKM+17], are also listed.

In the following, all the masses and their discrepancies will be discussedin detail. The TOF-ICR spectra for all the nuclides are listed in Appendix B.

Table 7.2: Frequency ratios between the ions of interest and reference ions.

Nuclide T1/2 Ref r =νc,refν ME (keV) ∆ME (keV)

ISOLTRAP AME2012140CeO? Stable 133Cs 1.173017796(19) −92809.0(2.3) −92816.2(2.2) 7.2(3.2)140NdO? 3.34 d 133Cs 1.173048610(26) −88994.3(3.2) −88991(26) −3(26)156Dy? Stable 133Cs 1.173197811(34) −70523.1(4.2) −70528.3(1.6) 5.2(4.5)160Yb? 4.8 m 133Cs 1.203394356(44) −58163.2(5.5) −58165(16) 2(17)168Lum 6.7 m 133Cs 1.263597908(46) −56908.2(5.8) −56870(39) −38(40)178Yb 74 m 85Rb 2.095672299(110) −49663.1(8.7) −49694(10) 31(13)52Cr? Stable 39K 1.333053034(12) −55419.7(0.4) −55418.1(0.6) −1.6(0.8)52Cr 85Rb 0.6116970566(67) −55419.4(0.5) −1.2(0.8)53Cr Stable 85Rb 0.6234757106(68) −55288.8(0.5) −55285.9(0.6) −2.9(0.6)54Cr Stable 85Rb 0.6352318839(81) −56936.4(0.6) −56933.7(0.6) −2.7(0.7)55Crb,?,R 3.497 m 85Rb 0.6470319500(225) −55112.3(1.8) −55108.6(0.6) −3.6(1.9)56Cr? 5.94 m 56Fe 1.000102145(9) −55284.4(0.7) −55281.2(1.9) −3.2(2.0)57Crb,?,R 21.1 s 85Rb 0.6706186690(233) −52525.0(1.8) −52524.1(1.9) −0.8(2.6)55Mn Stable 85Rb 0.646999088(9) −57711.5(0.7) −57711.7(0.4) 0.2(0.9)55Mnb,? 85Rb 0.6469990510(225) −57714.4(1.8) −2.7(1.8)59Fe? 44.495 d 85Rb 0.694069772(13) −60664.1(1.0) −60664.2(0.5) 0.1(1.1)75GaR 126 s 85Rb 0.882403259(9) −68460.6(0.7) −68464.6(2.4) 4.0(2.5)77Ga 13.2 s 85Rb 0.905988440(53) −65995.0(4.2) −65992.3(2.4) −2.6(4.9)78Ga 5.09 s 85Rb 0.917794409(14) −63704.0(1.1) −63706.0(1.9) 2.0(2.2)79Ga 2.848 s 85Rb 0.929586018(20) −62548.8(1.6) −62547.7(1.9) −1.1(2.5)

m Assigned to the Jπ = 3+ isomeric state.R Measured by Ramsey’s method.b Extra systematic error was added to account for helium-filling.? Already included in AME2016.

168Lu

Two isomers were reported in 168Lu [CCP+72] with half-lives of 5.5 min(Jπ = 6−) and 6.7 min (Jπ = 3+). In that experiment, the authors obtainedtwo distinct β+ spectra. The excitation energy for the higher isomeric state

99


(a) TOF-ICR spectrum using a two-pulse Ramsey scheme with two 100 ms durationtimes and 1 s waiting time.

(b) TOF-ICR spectrum with excitation time Trf = 1.2 s.

Figure 7.5: Time-of-flight spectrum of 57Cr.

100


(Jπ = 3+) was determined to be 220(130) keV from two endpoint energies. In1997, the level scheme of 168Lu was re-investigated [BATH97]. The intensityof a γ transition of 202.8 keV was determined to be 0.86(0.21) per 100 decays,which is much lower than the prediction [CCP+72]. The authors in [BATH97]concluded that: “This transition is very weak in the γ channel: it was onlyobserved in the total sum spectrum of all irradiations, so it was considereddoubtful ...” No other transition from a low-spin state in the neighborhood of220(130) keV was reported. This γ-ray might be interpreted as the transitionbetween the two isomeric state.

The production yields of the rare-earth nuclides were studied at ISOLTRAP[BAA+00]. By the measurements of the γ-line intensities, the relative produc-tion ratio between the isomeric state and the ground state for 168Lu was deter-mined to be ∼ 20 : 1 (see Table 7 in [BAA+00]). However, no mass informationconcerting 168Lu was given. The only direct mass measurement of 168Lu wasperformed using Schottky mass spectrometry [LGR+05] at GSI with precisionof 28 keV, while no excited isomeric state was reported.

In the current study, we tried to produce two states in 168Lu using thebombardment of proton beams on a tantalum target, the same productionmechanism as in [BAA+00], and measure the masses of the two states directlyat ISOLTRAP. Fig. 7.6a shows the TOF-ICR spectrum of 168Lu with excitationtime Trf = 1.2 s. We notice that there exists only one resonance. To separatetwo states which differ by ∼ 200 keV, the minimum resolving power Rmin =168× 931.494× 1000/200 ∼ 7.8× 105 is required. The resolving power of theprecision Penning trap at Trf = 1.2 s is R = 1.25× 540607× 1.2 ≈ 8.1× 105.Since the resolving power of the current setting is merely larger than Rmin,the two states could be mixed together.

To avoid the possibility of mixing two resonances, we increased the exci-tation time to Trf = 3 s, which would increase as a consequence the resolvingpower by a factor of 2.5. The TOF-ICR spectra at Trf = 3 s is shown inFig. 7.6b. One notices that the width of the resonance at Trf = 3 s did notreduce significantly. It was due to the collisions between the ions and the restgas atoms, as the ions were stored for a longer time. It is clear that, eventhough the rest gas might play a role, only one resonance was found and notrace of a second minimum TOF was seen.

Based on the previous study [BAA+00], where the isomeric state of 168Luwas populated a factor of 20 higher than the ground state, the resonance peakin Fig. 7.6 is assigned to the isomeric state. The arrows in Fig. 7.6 indicatethe position of the expected ground state, if produced. Based on the fact thatno second resonance is visible, we are convinced that only the isomeric statewas produced in the experiments. If the ground state were produced, as theexcitation time was increased to Trf = 3 s, a second peak would have appearedat the corresponding position indicated by the arrows in Fig. 7.6.

The mass for the 168Lu Jπ = 3+ isomeric state is determined to be−56908.2(5.8) keV, which is in agreement with the result of −56922(28) keV

101


(a) (b)

Figure 7.6: TOF-ICR spectra for 168Lu. Two excitation times are taken: Trf =1.2 s (a) and Trf = 3 s (b). The arrows indicate the position of the expectedground state.

from the Schottky measurement [LGR+05], where 168Lu was produced by thefragmentation of a bismuth target and its mass was assigned to the groundstate. In AME2012, it was assumed that the result from the Schottky measure-ment could suffer from the mixture of two isomeric states that its value wascorrected by evaluators to −57023(65) keV for the ground state. Combiningthe isomeric-state mass of 168Lu in the current analysis with the mass valuefor the ground state in AME2012 yields the excitation energy of the isomericstate of 160(40) keV, which is compatible with the excitation energy deducedfrom the two endpoint energies [CCP+72]

178Yb178Yb is the last known nuclide in the ytterbium isotopic chain. Its mass

was determined from a 176Yb(t,p) reaction [ZBM+82] with a precision of10 keV. The new ISOLTRAP value shows a difference of 31(13) keV from thereaction value. As mentioned before, the mass derived from a reaction dependsalso on the other masses. Sometimes, the recalibration could change a massby around 20 keV (the reaction Q-value will not change) if new masses areused for the nuclides involved in the reaction. In [ZBM+82], the reaction wascalibrated using two reference reactions 12C(t,p)14C and 16O(t,p)18O. Sincethe masses of all the involved nuclides were well known at that time, such adifference cannot be due to recalibration.

Fig. 7.7 displays the two-neutron separation energy in the ytterbium re-gion. For isotopic chains with high Z number Hf (Z = 72), Ta (Z = 73), andW (Z = 74), the regular behavior of S2n breaks down at N = 108, which wasinterpreted in terms of an energy gap above the Nilsson single-particle level92

+[624] [BMS+73]. Above N = 107, the S2n value at 178Yb (Z = 70) remains

102


Figure 7.7: Two-neutron separation energy in the ytterbium region between(Ho (Z = 67) and W (Z = 74)). The experimental data are denoted by blackcircles, estimated masses are denoted by empty diamonds, and the red circlerepresents the new 178Yb mass.

almost unchanged, which means extra binding is gained in 178Yb. Even thoughthe new ISOLTRAP result for 178Yb differs from the reaction value [ZBM+82]by 31(13) keV, the extra binding energy of ∼ 440 keV is confirmed. A sud-den flattening of S2n could be explained as Quantum Phase Transition (QPT)in atomic nuclei [Cas09]. For example, the discontinuity of S2n at N ∼ 90in 60Nd, 62Sm, and 64Gd [DSI80] signals the spherical-deformed transition re-gion. And the transition phenomenon can also be seen as a striking change ofR4/2 ≡ E(4+

1 )/E(2+1 ) for the 60Nd, 62Sm, and 64Gd isotopic chains (see Fig. 3

of [CWBG81]), where E(4+1 ) and E(2+

1 ) are the excitation energies of the first4+ and 2+ states, respectively. In the current mass region, 176Tm (Z = 69) isthe last known nuclide of thulium. Its mass came from a β− decay [67Gu11]with poor precision of 100 keV. The S2n at 176Tm decreases significantly com-pared to other nuclides at N = 107. This nuclide is poorly known, it couldbe an excited isomeric state. In lower Z region, neither mass information norspectroscopic data is available after N = 105. To clarify if there exists also aphase transition in the ytterbium region at N = 108, more experiments arecalled for in this mass region.

103


Other rare-earth masses

In AME2012, the mass of 140Ce was determined by several methods. TheISOLTRAP result of 140Ce (cerium oxide) differs from the value in AME2012by 7.2(3.2) keV. However, this mass was included in AME2016, and the globaladjustment shows that it agrees with the adjusted value within 1.2σ.

The mass of 140Nd (neodymium oxide) agrees well with the previousSchottky measurement [LGR+05]. And the precision is improved by a factorof 8.

In AME2012, the mass of 156Dy was mainly determined by Penning trapspectrometry at SHIPTRAP [EGB+11]. The ISOLTRAP result agrees withtheir value within 1.2σ.

The mass of 160Yb was previously measured by ISOLTRAP [BAA+01].The new result agrees with the previous one perfectly and the precision isimproved by a factor of almost three.

Chromium masses

Table 7.3: Influences of the ISOLTRAP results and the adjusted chromiummasses.

Nuclide ISOLTRAP Influence % Adjusted Mass v/s52Cr −55419.7(0.4) 27 −55419.82(0.23) 0.3

−55419.4(0.5) 21 −0.853Cr −55288.8(0.5) 24 −55287.68(0.23) 2.554Cr −56936.4(0.6) 18 −56935.44(0.24) 1.655Cr −55112.3(1.8) 0 −55110.39(0.30) 1.1

The masses of 52−57Cr were measured in two experimental campaigns in2014 and 2016. The results show that the determined masses of the chromiumisotopes are systematically smaller than the values in AME2012 (see Ta-ble 7.2). In AME2012, the masses from 52Cr to 55Cr were mainly determinedby a series of (n,γ) reactions with precision better than 0.3 keV. A (p,γ) reac-tion also plays a role in the determination of the mass of 54Cr. Other reactionscontribute much less to the chromium mass region under discussion. As boththe Penning trap measurements and the reaction measurements have compa-rable precision, the origin of these differences remains unclear. In this case,to quantify the influences of the new measurements on the existing chromiummasses, all the chromium results were included in the global adjustment. Theinfluences of the ISOLTRAP results and the adjusted masses of 52−55Cr arelisted in the third and the fourth column of Table 7.3, followed by v/s in thefifth column. The results from the global adjustment shows that the masses

104


of 52Cr, 54Cr, and 55Cr in the current analysis agree with the adjusted masseswithin 1.6σ, while the mass of 53Cr differs from the adjusted one by 2.5σ.

Fig. 7.8 displays the flow of information diagram in the chromium region,where the numbers in black represents the evaluation outcome in AME2012,while the numbers in blue represent the new evaluation results. We can seefrom Fig. 7.8 that the mass of 53Cr is now determined by 52Cr(n,γ) (42.9%),53Cr(n,γ) (32.6%) and the ISOLTRAP result (24.5%). The discrepancy orig-inates from the mass difference between 52Cr and 53Cr. The Q value for52Cr(n,γ)53Cr is determined to be 7940.5(0.6) keV based on the ISOLTRAPresults. However, this value is not consistent with the three input values7939.5(0.3) keV [IKKP80], 7939.0(0.2) keV [KCL80] and 7939.1(0.3) keV [INT07],which give the average value 7939.15(0.14) keV. The difference between theISOLTRAP result and the average input value is 1.6(0.6) keV.

Using the masses of 53Cr and 54Cr, theQ value for the reaction 53Cr(n,γ)54Crcan be derived to be 9718.9(0.7) keV. In AME2012, this reaction Q value wasthe average of four results † 9719.3(0.2) keV [WGB68], 9718.3(0.4) keV [LT72],9718.9(0.3) keV [IKKP80], and 9719.7(0.5) keV [Hof89], which gives the av-erage value of 9719.14(0.13) keV. The ISOLTRAP result for the reaction Qvalue of 53Cr(n,γ)54Cr is in agreement with the four average values.

The mass of 53Cr was measured in the same run as 52Cr, 54Cr, and 55Mn.The v/s values for the later three nuclides are within 1.5σ with respect tothe new adjustment. We found no reason for the discrepancy of 53Cr. For themoment, the Penning-trap mass of 53Cr and the three 52Cr(n,γ) reaction Qvalues are used to determine the mass of 53Cr. Remeasurements of the massof 53Cr is highly desired to clarify this discrepancy.

Including all the ISOLTRAP results for the chromium masses in the globaladjustment, the precision of all the masses of 52−55Cr has been improved bya factor of two, which is indicated in the lower-right corner of each box inFig. 7.8.

The masses of 56Cr and 57Cr were measured previously by ISOLTRAP[GAB+05]. The new ISOLTRAP results differ from the previous ones by−3.2(2.0) keV and −0.8(2.6) keV, respectively. The results obtained from thecurrent analysis agree well with the old results but the uncertainty of the massof 56Cr is improved by a factor of three.

Gallium masses

The masses of 75,77−79Ga were measured previously by ISOLTRAP [07Gu09].The new measurements for these gallium isotopes agree with the previous re-sults within 1.6σ. The precision for 75Ga is three times higher than the preciousresult.

†. The original values were recalibrated by Wapstra.

105


Figure 7.8: Flow of information diagram for the chromium masses from A = 52to A = 55. Each box represents a nuclide, with the mass uncertainty (in keV)in the lower right corner. The numbers in black represent the old evaluationin AME2012, and numbers in blue represent the new evaluation including thenew chromium results. The numbers in blue in the lower parts indicate theinfluences of the current data on the corresponding nuclides. The dash arrowsindicate the contribution from other experiments.

55Mn and 59Fe

The mass of 55Mn was well-known in AME2012 with a precision of 0.4 keV.The two measurements of 55Mn in 2014 and 2016 differ from the AME2012value by −2.7(1.8) keV and 0.2(0.9) keV, respectively, within 1.5σ.

The mass of 59Fe was determined to be −60664.2(0.5) keV in AME2012by (n,γ) reactions. Our value of −60664.1(1.0) agrees perfectly with the rec-ommended values.

All the results in this thesis have been included in the AME adjustment.We can obtain the v/s values for 16 out of 20 cases: eight cases smaller thanone, seven cases between one and two, and one case between two and three.The reduced chi-square is determined to be 1.2, to which the discrepancy of53Cr contributes the most.

106

Chapter 8

Systematic error studies of theMR-TOF MS at ISOLTRAP

8.1 Principle of MR-TOF MSThe MR-TOF device serves as a versatile tool for mass determination.

It can be used as a mass separator before the preparation Penning trap, andmost importantly, it can be used as a mass spectrometer. For nuclides withhalf-lives below 100 ms, the MR-TOF MS is superior to the Penning trapspectrometry with its higher resolving power [WWA+13].

Ions with different mass-to-charge ratio (m/q) can be separated from eachother longitudinally if they are accelerated by the same potential U , from whichthe ions acquire kinetic energy:

Ek = qU = mv2/2. (8.1)

The MR-TOF MS can be tuned in such a way that the time of flights areindependent of the energies of the ions and depend only on their mass-over-charge ratios (isochronicity). The time of flights, after passing through thesame length, are mass dependent: t ∝ 1/v ∝

√m, assuming q = 1.

The relation between the mass-to-charge ratio and the TOF can be de-termined as follows:

t = α(m/q)1/2 + β, (8.2)

where α and β are two parameters related to the MR-TOF device. The TOFspectrum can be transformed into a mass spectrum by measuring the TOFsof two reference ions with well-known masses:

t1 = α(m1/q)1/2 + β, t2 = α(m2/q)

1/2 + β, (8.3)

from which α and β can be determined.We introduce a parameter [WBB+13a]:

107

8.2. SYSTEMATIC ERROR STUDY

CTOF = (2t− t1 − t2)/2(t1 − t2), (8.4)

where t, t1 and t2 are the TOFs of the ion of interest and the two referenceions, respectively. Then the unknown mass is:

m1/2 = CTOF∆ref + Σref/2, (8.5)

where ∆ref = m1/21 −m1/2

2 and Σref = m1/21 +m

1/22 . The uncertainty of CTOF

is:

∆CTOF =√

∆t2/(t2 − t1)2 + (t− t2)2/(t2 − t1)4∆t21 + (t− t1)2/(t2 − t1)4∆t22.

(8.6)

8.2 Systematic error studySystematic errors exist in all experimental devices and can be significant

or not. Unlike random errors, which can be reduced by repeating the measure-ments, the systematic bias can not be eliminated by simple repetition and canaffect the accuracy of a measurement.

In this thesis, an off-line ion source was used to study the systematic errorsin MR-TOF MS. The off-line reference ions are first decelerated, accumulated,and bunched in the RFQ buncher, and then re-accelerated to Ek ≈ 3 keV(Fig. 7.4). Since the energy of the ions is larger than the maximum potential ofthe entrance mirror electrode, they will pass this electrode and enter into drifttube. At this time, an in-trap lift [WMRS12] voltage Ulift of 1 kV is appliedto the drift tube, which reduces the energy of the ions to Etrap = Ek − eUlift.When the ions pass through the center of the lifted electrode, this electrodeis switched to ground. Since the energy of the ions is no longer larger thanthe maxima of the two mirror electrode potentials, the ions are trapped andbounce back and forth between the two mirror electrodes. After a few hundredreflections, the in-trap potential is lifted again. The ions regain the energy andpass the exit mirror electrode, and are detected by the MCP. The signals fromthe MCP are registered by a multiple-event time digitizer (MSC6A).

In the following, different settings of the MR-TOF MS will be discussed.

Off-line Study

Setting 1: Measurement of 39,41K, 85,87Rb, 133Cs startingfrom reflection number 100 to 1000.

In this setting, five reference ion species 39,41K, 85,87Rb, 133Cs were usedand their TOFs were measured sequentially from reflection 100 to 1000 with

108

CHAPTER 8. MR-TOF MS

steps of 100 reflection. For each species, the TOFs were fitted as a function ofthe number of reflection using a linear model:

T = aN + b, (8.7)

where a and b are two mass-dependent parameters.

Figure 8.1: Time of flight of five reference species fitted as a function of thenumber of reflection.

Figure 8.1 shows that the TOFs are proportional to the number of reflec-tion for each species. Table 8.1 lists the fitting parameters a and b. Obtainingthese two parameters, we can calculate the TOF at any number of reflection.In order to see how the TOF differs from the linear trend, the relative residual,which is defined as the difference between “Raw TOF” (obtained from experi-ments) and the TOF extracted from Eq. 8.7 (“Fit” TOF) divided by the RawTOF, is also plotted in Fig. 8.2. As one can see, for relatively small reflectionnumber, the relative residuals are around 5 × 10−6, which was probably dueto the imperfect injection and the saturation of the detector. While as thenumber of reflection increases, the unstable effects can be averaged out andthe relative residuals can be reduced below 1× 10−6.

The TOFs at N = 0 (ions shooting through the MR-TOF MS withouttrapping) for five species can be also fitted by a linear function:

T (N = 0) = b = a0

√m+ b0, (8.8)

109


ion a (ns) a error (ns) b (ns) b error (ns)39K 15016.044 0.010 23165.548 6.47541K 15396.275 0.005 23720.824 3.303

85Rb 22167.192 0.019 33969.901 13.45987Rb 22426.429 0.012 34340.019 8.384133Cs 27733.159 0.017 42332.383 11.773

Table 8.1: Fitting parameters a and b in Eq. 8.7

Figure 8.2: Illustration of the relative residuals as a function of reflection num-ber.

110


(a)

(b)

Figure 8.3: Relation between the square root of mass and the TOF at N = 0(a). Residuals of TOF at N = 0 (b).

111


where a0 and b0 are two mass-independent parameters in this case.

Fig. 8.3a shows the TOF at N = 0 as a function of square-root of mass.We observe that the data points are well described by the linear function inEq. 8.8. The two fitting parameters are deduced to be a0 = 3630.8(3.6) andb0 = 487.7(25.8) ns. The parameter b0 is probably due to the unsynchronizedtime signal sent to the RFQ for injection and to the MCP for data acquisi-tion. It confirms the necessity of the use of the offset parameter β in massdetermination in Eq. 8.2.

The mass of 87Rb was used as the ion of interest and its mass was de-rived for each number of reflection using 39K and 133Cs as two reference ions.The mass was determined using two sets of TOFs: the “Raw” TOF from theexperiment data and the “Fit” TOF (see above) from the linear fit function.The results are displayed in Fig. 8.4. We can see that the “Fit” masses followsa smooth trend and have a large uncertainty. It is because the fluctuationwas averaged in the linear fit function. And the masses from “Raw” TOFscatter around the “Fit” masses. The largest deviation of “Raw” TOF fromAME2016 is 1147(277) keV at N = 100, while the “Fit” TOF gives a deviationof 381(689) keV. At N = 1000, which was the normal setting in MR-TOF MS,the “Raw” TOF deviated by −75(12) keV and the “’Fit" TOF by 12(48) keV.

Figure 8.4: Mass of 87Rb determined by using 39K and 133Cs as referencemasses.

112


Setting 2: Many measurements of 85Rb, 87Rb and 133Csfrom reflection number of 50 to 750.

In this setting, the TOF spectra were obtained from the accumulation of100 single spectrum for each species from reflection number 50 to 750, by stepsof 50 reflections.

We define here the standard deviation for each reflection number:

std =

√∑Ki=1(ti − t)2

K − 1, (8.9)

where ti is the TOF for the i-th spectrum, t is the weighted mean of TOF andK is the number of the spectrum (K = 100 for all reflection number exceptK = 86 for N = 750). The calculated standard deviation for each reflectionnumber is shown in Fig. 8.5. We can see that the std generally increases withthe reflection number, it is probably due to the fact that the cavity voltagewas not optimal and the deviations were somehow accumulating. The rising ofthe standard deviation at low reflection is again due to the imperfect injectionand saturation.

The relative residuals (see Fig. 8.6) defined in Setting 1 was also calculatedfor this setting. Fig. 8.6 shows that the relative residual of TOFs of threespecies are strongly related to each other. It is because the time of flights ofthree species were measured continuously that they suffered from the sameelectric field drift.

The mass determination of 87Rb is shown in Fig. 8.7. We can see thatthe deviation is largely suppressed when we used the TOF from the averageof 100 spectra. The deviations at N = 100 are 259(25) keV and 164(385) keVfor the “Raw” TOF and the “Fit” TOF, respectively. Above reflection numberN = 500, the deviation is less than 25 keV.

In order to see how the deviation changes as a function of the number ofthe accumulated spectra, the mass of 87Rb was calculated for different numberof accumulated spectra. We started with 10 spectra and increased by 10 eachtime, and finally the total number of 100 spectra for each reflection numberwas used. The results are displayed in Fig. 8.8. In this setting, the mass de-viation for N = 100 increases continuously as a function of the number ofmeasurement, while the deviations for other reflection numbers slowly con-verge at larger number of spectra. Fig. 8.8 gives us the first impression ofhow the deviation changes as a function of the number of the accumulatedspectra. If we examine carefully Fig. 8.8, we can notice complex structure fordifferent reflection numbers, i.e., the position of the minimum deviation is notthe same. For example, for N = 300 (the green curve), its minimum deviationappears at the number of spectra of 60; for N = 600 (the brown curve), its

113


Figure 8.5: Standard deviation of 100 TOF spectra for each species at differentnumber of reflection.

Figure 8.6: Relative residuals for three reference ions. The data was averagedby 100 spectra.

114


Figure 8.7: Mass determination of 87Rb from 100 spectra using 85Rb and 133Csas reference ions.

Figure 8.8: Mass of 87Rb determined by varying the number of the accumulatedspectra at different reflection number.

115


minimum deviation is found at the number of spectra of 100. We can con-clude from Fig. 8.8 that the measurements performed at reflection numberN = 100 is not reliable, and the minimum deviation is found at reflectionnumbers of N = 600 and N = 700 (in this setting) for the largest number ofmeasurements.

The mass-independent parameter b0 in this setting is determined to be683.2(9.7) ns, which is not too far away from the value determined from Set-ting 1.

Setting 3: Adjustment of the cavity voltage for each re-flection number.

As mentioned in the principle at the beginning of this chaper, the velocityof ions in the MR-TOF are modulated by the in-trap voltage Ulift. One canmodify the kinetic energy of the ions, i.e., by changing the value of Ulift, asillustrated in Fig. 3 of [WWA+13], to focus the time-of-flight plane of the ionson the detector. With such adjustment, the dispersion of the time-of-flightspectrum can reach minimum. Fig. 8.9 shows a typical Ulift voltage scan of85Rb at reflection number N = 900 from 1000 V to 1080 V in steps of 4 V.As one can see the TOFs, in the lower part of Fig. 8.9, varies at different Uliftvalues (represented as steps in y-axis). The minimum width of TOF locatesat Ulift = 1056V . The beam intensity, shown in the upper part of Fig. 8.9,remains almost unchanged.

In this setting, the cavity voltage was scanned for each number of reflec-tion to make sure that the dispersion of TOF is minimum. The correspondingcavity voltage is displayed in Fig. 8.10.

The measurements were performed at the corresponding optimum volt-ages. Two data sets for each species were acquired: one resulted from 10 spectrafor each number of reflection while the other from 40 spectra.

Fig. 8.11 shows the mass of 87Rb. We can see that the deviation, afteradjusting the cavity voltage, is strongly compressed, especially at the lowerreflection number, comparing with the red points in Fig. 8.7. Remember thatin the current setting, only 10 and 40 spectra were taken while in the Setting3 a hundred spectra were accumulated. By only adjusting the cavity voltage,the deviation for both measurements can be greatly reduced. The deviation ofthese two settings at reflection number of N = 1000 is 43(6) keV for 10 accu-mulated spectra and 0.1(3.0) keV for 40 accumulated spectra.

Setting 4: Delayed measurements at N = 1000.

In this setting, the reflection number was fixed at N = 1000 for threereference species 85Rb, 87Rb and 133Cs. We also introduced a delayed time indata acquisition: after each five measurements, an increment of 30 s was addedto each measurement. See below a sequence example:

116


Figure 8.9: Scan of Ulift voltages of 85Rb at N = 900 from 1000 V to 1080 Vin steps of 4 V (lower figure). Beam intensity as a function of Ulift.

Figure 8.10: Optimized cavity voltage as a function of reflection number N of85Rb

117


Figure 8.11: Mass determination of 87Rb at the optimum cavity voltage fortwo different data set.

85Rb – 30s – 87Rb – 30s – 133Cs – 30s85Rb – 30s – 87Rb – 30s – 133Cs – 30s85Rb – 30s – 87Rb – 30s – 133Cs – 30s85Rb – 30s – 87Rb – 30s – 133Cs – 30s85Rb – 30s – 87Rb – 30s – 133Cs – 30s85Rb – 60s – 87Rb – 60s – 133Cs – 60s

. . .

It means that the spectra were not taken at the same time (at least not in arelatively short time). The intention of this setting is to examine how deviationchanges along with the time difference between two measurements increases.In this setting, we obtained 72 spectra from 20:00 p.m. to 09:30 a.m. the nextday, the biggest time difference between two spectra was about 7 minutes. Theresults are listed in Fig. 8.12a. From Fig. 8.12, we can see that the deviationsdoes not depend on whether the spectra were taken in a fast mode betweentwo ions or not: for the very first spectra, they were measured promptly; forthe last spectrum, the time difference between the measurements of two ionswas seven minutes. It could imply that deviation can not be accounted for thedrift of the electrode voltage: an undiscovered source should be responsiblefor this deviation. For a single spectrum, a deviation around 500 keV can bereached. This is not a surprising result since each ion was measured at differenttime (They would never appear at the same spectrum for off-line ions). It thusposed an upper limit of 500 keV for the mass accuracy.

118


As the deviation is much larger than the uncertainty, the calculation of v/sdoes not make sense in this case. Thus all the measurements can be consideredindependent. The dispersion of the deviations is displayed in Fig. 8.12b. AGaussian fit is applied to the distribution of deviation, which gives a meanvalue of 17 keV and a standard deviation of 170 keV. If we use off-line beamsas reference ions, a systematic error of 170 keV should be added quadraticallyto the statistical uncertainties.

Conclusion (off-line ion source)

– The parameter β in Eq. 8.2 is mass independent and has a value around500 ns. It is probably due to the time difference between the ion injec-tion into the RFQ trap and the trigger sent to the acquisition card. Thedelay of the ion injection could minimize this quantity.

– The mass calculated from the “Fit” TOF agrees better than that fromthe “Raw” TOF. It is due to the fact the the fluctuation was averagedfor each reflection number.

– Measurements performed at reflection number N = 100 is not reliable,and the minimum deviation is found at reflection numbers of N = 600and N = 700 (in Setting 2) for the largest number of measurements.

– The deviation is probably not due to the drift of the electrode voltage,see Fig. 8.12. A yet undiscovered reason should be responsible for thisdeviation.

– In one single measurement (spectrum), a maximum deviation of 500 keVcan be observed. It thus set the upper limit for the mass precision. Thedispersion for one single measurement is 170 keV.

On-line Study

Up to now, the discussion is based on the off-line ion sources. Recently,measurements have been performed to check the yield of exotic species usingproton beams in various target. This provides a good opportunity to studythe systematic errors of MR-TOF during on-line runs.

The yield check measurements were carried out for nuclides with massnumber varying from A = 46 to A = 175. However, not all the data were usedin the systematic analysis since the measurements were not optimized for massmeasurements. Below lists all the criteria in the data selection:

– the beam gate was set properly so that no saturation was seen in thepeak.

– a spectrum contained at least two peaks.– a peak was identified without contaminants.– the uncertainty of the time of flight for each peak should be smallerthan 12 ns both for ions of interest and reference ions.

119


(a)

(b)

Figure 8.12: (a) Mass of 87Rb determined at reflection number 1000. The x-axisindicates when the measurement was performed. (b) Distribution of deviationsof (a). The red curve represents a Gaussian fit.

120


Figure 8.13: Time-of-flight spectrum of A = 46 nuclides.

– settings were not changed during data acquisition, e.g. no buncher cool-ing scanning, laser on/off tests, etc.

Fig. 8.13 shows a TOF spectrum of A = 46 nuclides that meets all thecriteria. In Fig. 8.13, 46Ca was selected as the ion of interest, 46Ti and 46Kwere used as references. The most abundant peak of 46Sc can also be used asthe ion of interest. However, its production strongly depends on the laser ionsource: very few scandium isotopes could be seen if the laser was turned off.This was indeed the aim for the yield check measurement!

In the mass determination process, the ions with the smallest error on theTOF were selected as the ions of interest and other ions were used as references.In total, under such selected criteria, 47 measurements which include 15 casesof A = 46, two cases of A = 48, nine cases of A = 49, five cases of A = 50,three cases of A = 62, six cases of A = 74, and seven cases of A = 149 wereused.

Fig. 8.14 illustrates the deviations between the masses determined fromexperiments and that from AME2016. The normalized chi-square, which isdefined as:

χn =

√1

N

∑i

(M iexp −M i

AME)2

σ2i

, (8.10)

is deduced to be 1.02 for all the 47 measurements under consideration, whereσi is the statistical uncertainty of the i-th mass. The expected interval of χnis 1± 1/

√(2 ∗ 47) = 1± 0.10, which is overlapped by the experimental value.

The systematic error derived from the on-line measurements is much smallerthan the statistical uncertainties, i.e., no systematic error should be added tothe final uncertainty.

121


The insert figure in Fig. 8.14 displays the distribution of v/s: 36 caseswithin 1σ, seven cases between 1σ and 2σ, three cases between 2σ and 3σ,and one case slightly larger than 3σ.

Figure 8.14: Mass deviations from AME2016 for 47 measurements. The insertfigure displays the distribution of the deviation divided by the uncertainty foreach measurement.

The effect of β in the mass calibration

For each of the 48 measurements, one could not always find two referenceions: the peak could be either contaminated by other component or weaklypopulated. In this case, only one reference peak was used in the mass cali-bration, meaning that the second parameter β in Eq. 8.2 was assumed to bezero.

Fig. 8.15 shows the β values when two references were used in the masscalibration. One would notice that its value, which was extracted from theon-line measurements, can be two orders of magnitude larger than that fromoff-line ion sources. However, considering the large uncertainty of each β value,we can assume that most of them are compatible with zero.

To examine the validity of using one reference ion in the mass calibration,one can do the following: compare the mass difference of the same nuclidedetermined by using one and two reference ions, respectively. The results are

122


Figure 8.15: Parameter β in Eq. 8.2 for on-line measurements.

displayed in Fig. 8.16. From Fig. 8.16 one can note that the difference isnegligible compared to the uncertainty. A linear fit to the data (including thelarge uncertainty in β) presents a very small value for the slope. In the currentstudy, the contribution of using one reference ion to the systematic error couldbe considered as negligible.

123


Figure 8.16: Difference of mass of the same nuclide determined by using oneand two reference ions. The red curve is the linear fit to the data point.

124

Conclusions and Outlook

In this thesis, the basic concept of Atomic Mass Evaluation (Ame) is de-scribed. Ame has been alive for more than 60 years and its main goal is toprovide all the information related to atomic masses, by evaluating all indi-rect (reaction and decay) and direct (mass spectrometry) mass measurements.However, since the masses are overdetermined by the experimental connec-tions, i.e., the number of measurements is much larger than that of masses,extracting the masses is not straightforward. Since all the input data is linearin mass, such an entangled system can be solved, without approximation, bythe least-squares method, where the masses are considered as parameters. Theuse of the least-squares method on the overdetermined system is an ideal pro-cedure in that it not only provides unbiased, reliable mass values derived fromexperimental data, but it also allows a check for the consistencies of all inputdata. One of the roles of Ame is to reveal undiscovered systematic errors bycomparing the input data with the adjusted values in a global prospect. Suchtask can only be performed under the Ame framework. After obtaining thebest value for each mass, we can calculate any combination of mass differences,such as decay and separation energies, based on the covariance matrix.

The developments for the latest mass table AME2016 are discussed. Thefirst one is related to the careful treatment of the most accurate data. The mostaccurate mass data comes from Penning-trap mass spectrometry. Nowadays, asthe precision from Penning traps can reach 10−10 or even better, the molecularand electronic binding energies cannot be neglected. A method to calculatethe molecular binding energy from the standard heat of formation is describedand two detailed examples are given. We find that, by using the updatedstandard heat of formation, the recalculated values for some of the mass-spectrometric data can change significantly. In AME2016, all the precise datahas been recalculated.

The second one is related to the corrections of decay energies. For α- andproton-decay energies measured by implantation methods, the recoil energyof the decaying partner should be taken into account properly. We presenta procedure to correct the published decay energies in case the recoiling nu-clides were not considered in implantation experiments. A program has beendeveloped based on Lindhard’s integral theory, which predicts accurately the

125


energy deposition of heavy nuclides in matter. Three examples are given toillustrate the correction procedure.

The third one is about the consideration of relativistic effect and atomiceffect in the formula of converting α- particle energies to decay energies. Themost precise α-decay energies come from magnetic spectrograph. These α-energy standards not only serve as calibration points for all α spectra withhigh resolving power, but also provide precise input values to Ame. In orderto obtain correct decay energies from magnetic spectrograph, where only αparticles are detected, a relativistic formula considering also the helium elec-tron binding energies is derived.

The mass models are the last resort to access to the most exotic nuclidesthat cannot be produced in the near future. These mass models, regardless oftheir intrinsic characters, could reproduce the experimentally known massesbut predict different behavior when they extrapolate towards unknown re-gions. Based on this fact, the accuracy and predictive power of eight massmodels of various types, i.e., Extended Thomas-Fermi plus Strutinsky Inte-gral method (ETFSI-2), Finite Range Droplet Model (FRDM95) and its up-dated version with improved treatment of deformation (FRDM12), a recentWeizsäcker-Skyrme plus Radial Basic Function (WS4+RBF) model, two re-cent Hartree-Fock-Bogoliubov mass models HFB26 and HFB27, the Duflo andZuker (DUZU) model, and the KTUY05 model, are studied. We find that theroot-mean-square deviation (δrms) of all the mass models under considerationare well below 0.8 MeV compared to three mass tables AME2003, AME2012,and AME2016 for all nuclides (N,Z ≥ 8). The mass model WS4+RBF is themost accurate mass model which gives δrms around 0.2 MeV. The microscopicmass model HFB27 also presents a good accuracy with δrms ≈ 0.5 MeV. Themass model DUZU is still a robust mass model which gives δrms ≈ 0.4 MeV.

The predictive power is the ability of a mass model to predict the un-known masses. To study the predictive power, we calculate the rms deviationsfor 61 newly measured masses (δrms(new)) in AME2016. To compare the pre-dictive power with the accuracy, the rms deviations for the known masses inAME2012 (δrms(2012)) are also calculated. In the Global region, the conditionrms(new)≈rms(2012) is only roughly fulfilled by the mass model ETFSI-2,while other mass models present a much large rms(new) value than that oftheir rms(2012). In the light region, ETFSI-2 has the best predictive power,while in other regions, the mass model WS4+RBF has the best predictivepower. In the heavy region, all the mass models find their lowest rms(new)value, and the predictive power is compatible with the accuracy.

The mass extrapolation of Ame provides the best estimates for the un-known masses that are not too far (two or three mass unit) from the lastknown nuclides. This method is based on the smoothness of the mass sur-face and such a smooth feature should be preferred when we extrapolate the

126


masses towards the unknown region. The root-mean-square deviation for allestimated masses in AME2012 that are known in AME2016 is 0.396 MeV (55cases in total), which is smaller than any of the the mass model discussed here.

Another part of this thesis is related to the mass measurements performedusing the Penning-trap mass spectrometer ISOLTRAP at ISOLDE/CERN.The masses of eighteen nuclides from several experimental campaigns between2011 and 2016 are analysed. The mass of 168Lu in its isomeric state has beenmeasured and its value agrees with the recommended value in AME2012 butnine times more precise. The mass value of 178Yb obtained here differs fromthe previous reaction result by 31(13) keV but confirms extra binding of 178Ybby ∼ 440 keV. A sudden flattening in S2n at 178Yb would indicate the existenceof the phase transition in the region. However, more measurements are neededto clarify this issue. The masses of some rare-earth nuclides such as 140Nd and160Yb are measured with higher precision compared to AME2016. The resultsof other nuclides also help improve the precision of the existing masses.

The systematic error of the multi-reflection time-of-flight mass spectrom-eter (MR-TOF MS) at ISOLTRAP is studied using off-line ion sources andon-line proton beams. In the off-line study, different settings on MR-TOF MShave been probed. Two of the most important conclusions are addressed. First,measurements performed at reflection number N = 100 is not reliable. In or-der to minimize the deviation, reflection numbers higher than 100 should beused instead. secondly, if we use off-line ion sources for mass calibration, anuncertainty of 170 keV should be added to the final result.

Secondly, the deviation is probably not due to the drift of the electrodevoltage. For the on-line study, 47 measurements are selected from the yield-check measurements. The reduced chi-square χn is deduced to be 1.02, meaningthat the systematic error is much smaller than the statistical uncertainties. Nosystematic error should be added to the final results. The effect of the secondparameter β in the mass determination formula is also studied. The resultshows that the use of only one reference ion will not affect for mass doubletsthe final determination of mass values and hence the conclusion of the system-atic error.

Ame serves as a reservoir which contains all the knowledge of atomicmasses. Unlike other fields of physics, whose aim is to prove something, theintention of Ame is to provide the physics community, by the best usage of theexisting data, with the best, reliable mass values with improved precision. Asa Ph.D student, I had the privilege to be one of the Ame evaluators. I also feelresponsible to keep Ame in good shape, after the retirement of my supervisorGeorges Audi. Ame is a project which takes a lot of time. The inclusion ofall experimental data is by no means a copy-paste task: it requires meticulousinspection and reconciles all the conflicting data. More importantly, evaluators

127


need to grasp the knowledge of all the experimental setup in order to have goodjudgement, which is also a long accumulation process. For the next two years,I will work at the Penning-trap mass spectrometer PENTATRAP installed atthe Max-Plank Institute for Nuclear Physics (MPIK) under the supervision ofProf. Klaus Blaum. It will certainly offer a great opportunity for me to gain anoverall knowledge of the Penning-trap technique and complement my presentskills of mass evaluation. In the future, I will strongly collaborate with Prof.Meng Wang, the coordinator of the Ame group at IMP, Lanzhou.

128

Appendix A

Relativistic formula of alpha decay

Lets consider that the parent nuclide of atomic mass M decays by theemission of an α particle or proton, denoted here as s, that has a nuclearmass ms and that the residual (daughter) nucleus has an atomic mass M ′

d (itincludes also two extract electrons for α-decay and one for proton decay).

Since the emission of a particle s is a two body process, the conservationof momentum requires:

~pp = ~ps + ~pd (A.1)

In the center-of-mass frame, the parent nuclide is at rest (~pp=0), so that ~ps =−~pd, absolute value of both is p. The total energy of the emitted particle andthe daughter nucleus can be expressed then as:

Etotal,s =√m2s + p2

Etotal,d =√M′2d + p2

(A.2)

where c = 1. The corresponding conservation of energy law requires that:

M = Etotal,s + Etotal,d =√m2s + p2 +

√M′2d + p2 (A.3)

and after re-arranging Eq. (A.3) one can obtain:

M2 = m2s + p2 +M

′2d +p2 + 2

√(m2

s + p2) · (M ′2d + p2)

(M2 −m2s −M

′2d − 2p2)2 = 4(m2

s + p2) · (M ′2d + p2)

M4 − 2M2 ·m2s − 2M2 ·M ′2

d −4M2 · p2 +m4s + 2m2

s ·M′2d + 4m2

s · p2+

+M′4d + 4M

′2d · p2 + 4p4 = 4(m2

s ·M′2d +m2

s · p2 +M′2d · p2 + p4)

M4 − 2M2 · (m2s +M

′2d ) + (m2

s −M′2d )2 = 4M2 · p2

(A.4)The momentum is then determined as:

p2 =M4 − 2M2 · (m2

s +M′2d ) + (m2

s −M′2d )2

4M2(A.5)

129

By substituting Eq. (A.5) into (A.2), for the total energy of the emitted particleone can obtain:

Etotal,s =

√4M2 ·m2

s +M4 − 2M2 · (m2s +M

′2d ) + (m2

s −M′2d )2

4M2

=

√M4 + 2M2 · (m2

s −M′2d ) + (m2

s −M′2d )2

4M2

=

√(M2 +m2

s −M′2d )2

4M2

=M2 +m2

s −M′2d

2M

(A.6)

The kinetic energy of the emitted particle is then determined as:

Es = Etotal,s −ms =M +ms −M

′2d

2M−ms =

(M −ms)2 −M ′2

d

2M(A.7)

The decay Q-value is defined as:

Qs = M −Ms −Md

= M − (ms + ns ·me −Be,s)− (M′

d − ns ·me)

= M −ms −M′

d −Be,s,

(A.8)

where ns is electron number (ns = 1 for proton decay and ns = 2 for α-decay)and me is the electron mass and Be,s is the electron binding energy of theemitted nuclide.

By re-arranging Eq. (A.8), one can obtain:

M′

d = M −ms −Qs −Be,s

= M −ms −Q′

s

(A.9)

whereQ′

s = Qs −Be,s (A.10)

and hence Eq. (A.7) becomes:

Es =(M2 −m2

s)2 − (M −ms −Q

′s)

2

2M

=2(M −ms) ·Q

′s −Q

′2s

2M

(A.11)

orQ′2s − 2(M −ms) ·Q

′

s − 2M · Es = 0 (A.12)

By solving Eq. (A.12), one can obtain:

Q′

s = (M −ms)±√

(M −ms)2 − 2M · Es (A.13)

130

APPENDIX A. RELATIVISTIC FORMULA OF ALPHA DECAY

Since, the solution with + sign before the square root does not have a physicalmeaning, one can write the expression for the Q value:

Q′

s = (M −ms)−√

(M −ms)2 − 2M · Es (A.14)

or alternatively:

Q′

s = (M −ms)− (M −ms) ·

√1− 2M · Es

(M −ms)2

= (M −ms) ·[1−

√1− 2M · Es

(M −ms)2

] (A.15)

Since 2M ·Es

(M−ms)2≈ 2Es

M≈ 15 MeV

1 GeV � 1, one can use the Binomial theorem:

√1 + x = 1 +

1

2x− 1

8x2 +

1

16x3 − · · · (A.16)

Thus, Eq. (A.15) can be rewritten as:

Q′

s = (M −ms) ·[1− 1 +

1

2

2M · Es(M −ms)2

+1

8

4M2 · E2s

(M −ms)4+ · · ·

]=

M

(M −ms)· Es +

1

2

M2

(M −ms)3· E2

s + · · ·

'[1 +

ms

(M −ms)

]· Es +

1

2

E2s

M

(A.17)

The first term in Eq. (A.17) is the non-relativistic part, while the second termis an estimate for the relativistic correction. Considering an α-decay of A = 200and Eα = 8 MeV, the relativistic correction at the first order would be:

1

2

EαMEα ≈

1

2

8

200 ∗ 931.494∗ 8 = 0.17 ∗ 10−3 MeV

or 0.17 keV.

Taking the electron binding energy into account, Eq. (A.14) can be rewrit-ten as:

Qs = (M −ms)±√

(M −ms)2 − 2M · Es +Be,s (A.18)

When the decay takes place between the ground-state of the parent andexcited state of the daughter nuclide, the decay Q-value should be revised to :

Qs = Q∗s − Ex (A.19)

where Ex is the excitation energy of the level.

131

Appendix B

TOF-ICR Spectra

(a) TOF-ICR resonance of 140CeO+ at Trf = 1200 ms.(b) TOF-ICR resonance of 140NdO+ at Trf = 1200 ms.

(c) TOF-ICR resonance of 156Dy+ at Trf = 1200 ms. (d) TOF-ICR resonance of 160Yb+ at Trf = 1200 ms.

133

(e) TOF-ICR resonance of 52Cr+ at Trf = 1200 ms. (f) TOF-ICR resonance of 53Cr+ at Trf = 1200 ms.

(g) TOF-ICR resonance of 54Cr+ at Trf = 1200 ms. (h) TOF-ICR resonance of 56Cr+ at Trf = 1200 ms.

(i) TOF-ICR resonance of 55Cr+ at Trf = 1200 ms. (j) Ramsey-type excitation TOF-ICR resonance of55Cr+ at (100− 1000− 100) ms.

134

APPENDIX B. TOF-ICR SPECTRA

(k) TOF-ICR resonance of 55Mn+ at Trf = 1200 ms. (l) Ramsey-type excitation TOF-ICR resonance of55Mn+ at (100− 1000− 100) ms.

(m) TOF-ICR resonance of 75Ga+ at Trf = 1200 ms. (n) Ramsey-type excitation TOF-ICR resonance of75Ga+ at (100− 1000− 100) ms.

(o) TOF-ICR resonance of 78Ga+ at Trf = 1200 ms. (p) TOF-ICR resonance of 79Ga+ at Trf = 1200 ms.

135

(q) TOF-ICR resonance of 59Fe+ at Trf = 1200 ms.

Figure B.-2: TOF-ICR spectra.

136

Bibliography

[ABB+11] V. N. Aseev, A. I. Belesev, A. I. Berlev, E. V. Geraskin, A. A.Golubev, N. A. Likhovid, V. M. Lobashev, A. A. Nozik, V. S.Pantuev, V. I. Parfenov, A. K. Skasyrskaya, F. V. Tkachov, andS. V. Zadorozhny. Upper limit on the electron antineutrino massfrom the troitsk experiment. Phys. Rev. D, 84:112003, Dec 2011.

[ADLW86] G. Audi, W. G. Davies, and G. E. Lee-Whiting. A method of de-termining the relative importance of particular data on selectedparameters in the least-squares analysis of experimental data.Nuclear Instruments and Methods in Physics Research SectionA: Accelerators, Spectrometers, Detectors and Associated Equip-ment, 249(2):443–450, 1986.

[AEH+10] A. N. Andreyev, J. Elseviers, M. Huyse, P. Van Duppen, S. An-talic, A. Barzakh, N. Bree, T. E. Cocolios, V. F. Comas,J. Diriken, D. Fedorov, V. Fedosseev, S. Franchoo, J. A. Here-dia, O. Ivanov, U. Köster, B. A. Marsh, K. Nishio, R. D. Page,N. Patronis, M. Seliverstov, I. Tsekhanovich, P. Van den Bergh,J. Van De Walle, M. Venhart, S. Vermote, M. Veselsky, C. Wage-mans, T. Ichikawa, A. Iwamoto, P. Möller, and A.J. Sierk. Newtype of asymmetric fission in proton-rich nuclei. Phys. Rev. Lett.,105:252502, Dec 2010.

[AKM+17] G. Audi, F. G. Kondev, W. Meng, W. J. Huang, and S. Naimi.The NUBASE2016 evaluation of nuclear properties. ChinesePhysics C, 41(3):030001, 2017.

[Arn96] W. D. Arnett. Supernovae and Nucleosynthesis : an Investiga-tion of the History of Matter, from the Big Bang to the Present.Princeton University Press, 1996.

[Aud01] G. Audi. The evaluation of atomic masses. Hyperfine Interac-tions, 132(1):7–34, Jan 2001.

[Aud06] G. Audi. The history of nuclidic masses and of their evalua-tion. International Journal of Mass Spectrometry, 251(2):85–94,2006. ULTRA-ACCURATE MASS SPECTROMETRY ANDRELATED TOPICS Dedicated to H.-J. Kluge on the occasionof his 65th birthday anniversary.

137

BIBLIOGRAPHY

[AW93] G. Audi and A. H. Wapstra. The 1993 atomic mass evaluation:(i) atomic mass table. Nuclear Physics A, 565(1):1 – 65, 1993.

[AWT03] G. Audi, A. H. Wapstra, and C. Thibault. The Ame2003 atomicmass evaluation: (ii). tables, graphs and references. NuclearPhysics A, 729(1):337 – 676, 2003. The 2003 NUBASE andAtomic Mass Evaluations.

[AWW+12] G. Audi, M. Wang, A. H. Wapstra, F.G. Kondev, M. Mac-Cormick, X. Xu, and B. Pfeiffer. The Ame2012 atomic massevaluation. Chinese Physics C, 36(12):1287, 2012.

[BA93] C. Borcea and G. Audi. New methods for extrapolating massesfar from stability. Rev. Roum. Phys, 38:455, 1993.

[BAA+00] D. Beck, F. Ames, G. Audi, G. Bollen, F. Herfurth, H. J.Kluge, A. Kohl, M. König, D. Lunney, I. Martel, R. B. Moore,H. Raimbault-Hartmann, E. Schark, S. Schwarz, M. de Saint Si-mon, and J. Szerypo. Accurate masses of unstable rare-earthisotopes by ISOLTRAP. The European Physical Journal A,8(3):307–329, Sep 2000.

[BAA+01] G. Bollen, F. Ames, G. Audi, D. Beck, J. Dilling, O. Engels,S. Henry, F. Herfurth, A. Kellerbauer, H.-J. Kluge, A. Kohl,E. Lamour, D. Lunney, R. B. Moore, M. Oinonen, C. Scheiden-berger, S. Schwarz, G. Sikler, J. Szerypo, and C. Weber. Massmeasurements on short-lived nuclides with ISOLTRAP. Hyper-fine Interactions, 132(1):213–220, Jan 2001.

[BAB+96] B. Blank, S. Andriamonje, F. Boué, S. Czajkowski, R. Del Moral,J. P. Dufour, A. Fleury, P. Pourre, M.S. Pravikoff, K.-H.Schmidt, E. Hanelt, and N. A. Orr. First spectroscopic study of22Si. Phys. Rev. C, 54:572–575, Aug 1996.

[BATH97] V. Barci, G. Ardisson, D. Trubert, and M. Hussonnois. Excitedstates in the doubly odd 168Lu nucleus fed by electron-capturedecay of 168Hf (T 1/2 = 25.95 min). Phys. Rev. C, 55:2279–2289,May 1997.

[BB36] H. A. Bethe and R. F. Bacher. Nuclear physics a. stationarystates of nuclei. Rev. Mod. Phys., 8:82–229, Apr 1936.

[BB08] B. Blank and M. J. G. Borge. Nuclear structure at the pro-ton drip line: Advances with nuclear decay studies. Progress inParticle and Nuclear Physics, 60(2):403 – 483, 2008.

[BBG+99] H. G. Bohlen, A. Blazevic, B. Gebauer, W. Von Oertzen,S. Thummerer, R. Kalpakchieva, S. M. Grimes, and T. N.Massey. Spectroscopy of exotic nuclei with multi-nucleon trans-fer reactions. Progress in Particle and Nuclear Physics, 42(Sup-plement C):17 – 26, 1999. Heavy Ion Collisions from Nuclear toQuark Matter.

138

BIBLIOGRAPHY

[BBH+02] K. Blaum, G. Bollen, F. Herfurth, A. Kellerbauer, H.-J. Kluge,M. Kuckein, E. Sauvan, C. Scheidenberger, and L. Schweikhard.Carbon clusters for absolute mass measurements at ISOLTRAP.The European Physical Journal A - Hadrons and Nuclei,15(1):245–248, Sep 2002.

[BDN13] K. Blaum, J. Dilling, and W. Nörtershäuser. Precision atomicphysics techniques for nuclear physics with radioactive beams.Physica Scripta, 2013(T152):014017, 2013.

[BFG+89] L. Bianchi, B. Fernandez, J. Gastebois, A. Gillibert, W. Mittig,and J. Barrette. Speg: An energy loss spectrometer for ganil.Nuclear Instruments and Methods in Physics Research SectionA: Accelerators, Spectrometers, Detectors and Associated Equip-ment, 276(3):509 – 520, 1989.

[BG86] L. S. Brown and G. Gabrielse. Geonium theory: Physics of asingle electron or ion in a Penning trap. Rev. Mod. Phys., 58:233–311, Jan 1986.

[BJP+91] V. Borrel, J. C. Jacmart, F. Pougheon, R. Anne, C. Detraz,D. Guillemaud-Mueller, A. C. Mueller, D. Bazin, R. Del Moral,J. P. Dufour, F. Hubert, M. S. Pravikoff, and E. Roeckl. 31Ar and27S: Beta-delayed two-proton emission and mass excess. NuclearPhysics A, 531(2):353 – 369, 1991.

[Bla06] K. Blaum. High-accuracy mass spectrometry with stored ions.Physics Reports, 425(1):1 – 78, 2006.

[BMS+73] R. C. Barber, J. O. Meredith, F. C. G. Southon, P. Williams,J. W. Barnard, K. Sharma, and H. E. Duckworth. Spacing ofnuclear ground-state levels in the region 94 ≤ n ≤ 114. Phys.Rev. Lett., 31:728–730, Sep 1973.

[BMSS90] G. Bollen, R. B. Moore, G. Savard, and H. Stolzenberg. Theaccuracy of heavy-ion mass measurements using time of flight-ion cyclotron resonance in a Penning trap. Journal of AppliedPhysics, 68(9):4355–4374, 1990.

[BND13] Y. Blumenfeld, T. Nilsson, and P. Van Duppen. Facilities andmethods for radioactive ion beam production. Physica Scripta,2013(T152):014023, 2013.

[BNW10] K. Blaum, Yu. N. Novikov, and G. Werth. Penning traps asa versatile tool for precise experiments in fundamental physics.Contemporary Physics, 51(2):149–175, 2010.

[Bol01] G. Bollen. Mass measurements of short-lived nuclides with iontraps. Nuclear Physics A, 693(1):3–18, 2001. Radioactive Nu-clear Beams.

[Bos77] K. Bos. Determination of atomic masses from experimental data.PhD Thesis, 1977.

139

BIBLIOGRAPHY

[BW72] K. Bos and A. H. Wapstra. Mass Extrapolation and Mass For-mulae, pages 273–277. Springer US, Boston, MA, 1972.

[Cas09] R. F. Casten. Quantum phase transitions and structural evo-lution in nuclei. Progress in Particle and Nuclear Physics,62(1):183 – 209, 2009.

[CCP+72] A. Charvet, R. Chéry, Do Huu Phuoc, R. Duffait, A. Emsallem,and G. Marguier. The decays of 5.5 min 168gLu and 6.7 min168mLu. Nuclear Physics A, 197(2):490 – 496, 1972.

[CS01] J. A. Cameron and B. Singh. Nuclear data sheets for A = 41.Nuclear Data Sheets, 94(3):429 – 603, 2001.

[CSS+04] J. A. Clark, G. Savard, K. S. Sharma, J. Vaz, J. C. Wang,Z. Zhou, A. Heinz, B. Blank, F. Buchinger, J. E. Crawford,S. Gulick, J. K. P. Lee, A. F. Levand, D. Seweryniak, G. D.Sprouse, and W. Trimble. Precise mass measurement of 68Se,a waiting-point nuclide along the rp process. Phys. Rev. Lett.,92:192501, May 2004.

[CW83] E. R. Cohen and A. H. Wapstra. Recommended treatmentof precision measurements related to nuclear energy levels andatomic masses. Nuclear Instruments and Methods in PhysicsResearch, 211(1):153 – 157, 1983.

[CWBG81] R. F. Casten, D. D. Warner, D. S. Brenner, and R. L. Gill.Relation between the Z = 64 shell closure and the onset of de-formation at N = 88− 90. Phys. Rev. Lett., 47:1433–1436, Nov1981.

[Dar70] B. D. Darwent. Bond dissociation energies in simple molecules.NSRDS-NBS NO. 31, U. S. DEPT. COMMERCE, WASHING-TON, D. C. JAN. 1970, 48 P, 1970.

[DG80] J. Dechargé and D. Gogny. Hartree-fock-bogolyubov calculationswith the d1 effective interaction on spherical nuclei. Phys. Rev.C, 21:1568–1593, Apr 1980.

[DMCD+03] P. De Marcillac, N. Coron, G. Dambier, J. Leblanc, and J. P.Moalic. Experimental detection of α-particles from the radioac-tive decay of natural bismuth. Nature, 422(6934):876–878, 2003.

[DNB+95] F. DiFilippo, V. Natarajan, M. Bradley, F. Palmer, and D. E.Pritchard. Accurate atomic mass measurements from Penningtrap mass comparisons of individual ions. Physica Scripta,1995(T59):144, 1995.

[DNBP94] F. DiFilippo, V. Natarajan, K. R. Boyce, and D. E. Pritchard.Accurate atomic masses for fundamental metrology. Phys. Rev.Lett., 73:1481–1484, Sep 1994.

140

BIBLIOGRAPHY

[DPB+15] T. Dickel, W. R. Plaß, A. Becker, U. Czok, H. Geissel, E. Haet-tner, C. Jesch, W. Kinsel, M. Petrick, C. Scheidenberger, A. Si-mon, and M. I. Yavor. A high-performance multiple-reflectiontime-of-flight mass spectrometer and isobar separator for the re-search with exotic nuclei. Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers, Detec-tors and Associated Equipment, 777:172 – 188, 2015.

[DSI80] A. E. L. Dieperink, O. Scholten, and F. Iachello. Classical limitof the interacting-boson model. Phys. Rev. Lett., 44:1747–1750,Jun 1980.

[DZ95] J. Duflo and A. P. Zuker. Microscopic mass formulas. Phys.Rev. C, 52:R23–R27, Jul 1995.

[DZ96] J. Duflo and A.P. Zuker. Private communication to G. Audi.February 1996.

[EBB+13] S. Eliseev, K. Blaum, M. Block, C. Droese, M. Goncharov,E. Minaya Ramirez, D. A. Nesterenko, Yu. N. Novikov, andL. Schweikhard. Phase-imaging ion-cyclotron-resonance mea-surements for short-lived nuclides. Phys. Rev. Lett., 110:082501,Feb 2013.

[EBB+15] S. Eliseev, K. Blaum, M. Block, S. Chenmarev, H. Dorrer, Ch. E.Düllmann, C. Enss, P. E. Filianin, L. Gastaldo, M. Goncharov,U. Köster, F. Lautenschläger, Yu. N. Novikov, A. Rischka, R. X.Schüssler, L. Schweikhard, and A. Türler. Direct measurementof the mass difference of 163Ho and 163Dy solves the q-valuepuzzle for the neutrino mass determination. Phys. Rev. Lett.,115:062501, Aug 2015.

[EGB+11] S. Eliseev, M. Goncharov, K. Blaum, M. Block, C. Droese,F. Herfurth, E. Minaya Ramirez, Yu. N. Novikov,L. Schweikhard, V. M. Shabaev, I. I. Tupitsyn, K. Zuber,and N. A. Zubova. Multiple-resonance phenomenon in neutri-noless double-electron capture. Phys. Rev. C, 84:012501, Jul2011.

[EKMW60a] F. Everling, L. A. König, J. H. E. Mattauch, and A. H. Wapstra.Atomic masses of nuclides for A ≤ 70. Nuclear Physics, 15:342– 355, 1960.

[EKMW60b] F. Everling, L.A. König, J. H. E. Mattauch, and A. H. Wapstra.Relative nuclidic masses. Nuclear Physics, 18:529 – 569, 1960.

[EKMW61] F. Everling, L. A. König, J. H. E. Mattauch, and A. H. Wap-stra. Adjustment of relative nuclidic masses (i) A ≤ 70. NuclearPhysics, 25(Supplement C):177 – 215, 1961.

141

BIBLIOGRAPHY

[FGM08] B. Franzke, H. Geissel, and G. Münzenberg. Mass and lifetimemeasurements of exotic nuclei in storage rings. Mass Spectrom-etry Reviews, 27(5):428–469, 2008.

[Fra87] B. Franzke. The heavy ion storage and cooler ring project esr atGSI. Nuclear Instruments and Methods in Physics Research Sec-tion B: Beam Interactions with Materials and Atoms, 24-25:18– 25, 1987.

[GAB+01] H. Geissel, F. Attallah, K. Beckert, F. Bosch, M. Falch,B. Franzke, M. Hausmann, Th. Kerscher, O. Klepper, H.-J. Kluge, C. Kozhuharov, Yu. Litvinov, K. E. G. Löbner,G. Münzenberg, N. Nankov, F. Nolden, Yu. Novikov, T. Oht-subo, Z. Patyk, T. Radon, C. Scheidenberger, J. Stadlmann,M. Steck, K. Sümmerer, H. Weick, and H. Wollnik. Progress inmass measurements of stored exotic nuclei at relativistic ener-gies. Nuclear Physics A, 685(1):115 – 126, 2001. Nucleus-NucleusCollisions 2000.

[GAB+05] C. Guénaut, G. Audi, D. Beck, K. Blaum, G. Bollen, P. De-lahaye, F. Herfurth, A. Kellerbauer, H.-J. Kluge, D. Lunney,S. Schwarz, L. Schweikhard, and C. Yazidjian. Mass measure-ments of 56−57Cr and the question of shell reincarnation atN = 32. Journal of Physics G: Nuclear and Particle Physics,31(10):S1765, 2005.

[GAB+07] C. Guénaut, G. Audi, D. Beck, K. Blaum, G. Bollen, P. De-lahaye, F. Herfurth, A. Kellerbauer, H.-J. Kluge, J. Libert,D. Lunney, S. Schwarz, L. Schweikhard, and C. Yazidjian. High-precision mass measurements of nickel, copper, and gallium iso-topes and the purported shell closure at n = 40. Phys. Rev. C,75:044303, Apr 2007.

[GBB+07] S. George, S. Baruah, B. Blank, K. Blaum, M. Breiten-feldt, U. Hager, F. Herfurth, A. Herlert, A. Kellerbauer, H.-J. Kluge, M. Kretzschmar, D. Lunney, R. Savreux, S. Schwarz,L. Schweikhard, and C. Yazidjian. Ramsey method of separatedoscillatory fields for high-precision Penning Trap Mass Spec-trometry. Phys. Rev. Lett., 98:162501, Apr 2007.

[GCP13a] S. Goriely, N. Chamel, and J. M. Pearson. Further explorationsof Skyrme-Hartree-Fock-Bogoliubov mass formulas. XIII. the2012 atomic mass evaluation and the symmetry coefficient. Phys.Rev. C, 88:024308, Aug 2013.

[GCP13b] S. Goriely, N. Chamel, and J. M. Pearson. Hartree-Fock-Bogoliubov nuclear mass model with 0.50 MeV accuracy basedon standard forms of Skyrme and pairing functionals. Phys. Rev.C, 88:061302, Dec 2013.

142

BIBLIOGRAPHY

[GCP16] S. Goriely, N. Chamel, and J. M. Pearson. Latest results ofSkyrme-Hartree-Fock-Bogoliubov mass formulas. Journal ofPhysics: Conference Series, 665(1):012038, 2016.

[GKT80] G. Gräff, H. Kalinowsky, and J. Traut. A direct determination ofthe proton electron mass ratio. Zeitschrift für Physik A Atomsand Nuclei, 297(1):35–39, Mar 1980.

[GMDN68] T. B. Grandy, W. J. McDonald, W. K. Dawson, and G. C. Neil-son. The 48Ca(d,n)49Sc reaction at Ed = 5.5 and 6.0 MeV. Nu-clear Physics A, 113(2):353 – 366, 1968.

[GMO+12] L. Gaudefroy, W. Mittig, N. A. Orr, S. Varet, M. Chartier,P. Roussel-Chomaz, J. P. Ebran, B. Fernández-Domínguez,G. Frémont, P. Gangnant, A. Gillibert, S. Grévy, J. F. Libin,V. A. Maslov, S. Paschalis, B. Pietras, Yu.-E. Penionzhkevich,C. Spitaels, and A. C. C. Villari. Direct mass measurements of19B, 22C, 29F, 31Ne, 34Na and other light exotic nuclei. Phys.Rev. Lett., 109:202503, Nov 2012.

[Gor00] S. Goriely. Nuclear inputs for astrophysics applications. AIPConference Proceedings, 529(1):287–294, 2000.

[GR71] B. Grennberg and A. Rytz. Absolute measurements of α-rayenergies. Metrologia, 7(2):65, 1971.

[HA17] W. J. Huang and G. Audi. Corrections of alpha- and proton-decay energies in implantation experiments. EPJ Web Conf.,146:10007, 2017.

[HAW+17] W. J. Huang, G. Audi, Meng Wang, F. G. Kondev, S. Naimi, andXing Xu. The ame2016 atomic mass evaluation (i). evaluationof input data; and adjustment procedures. Chinese Physics C,41(3):030002, 2017.

[HBGU+06] P. A. Hausladen, J. R. Beene, A. Galindo-Uribarri,Y. Larochelle, J. F. Liang, P. E. Mueller, D. Shapira,D. W. Stracener, J. Thomas, R. L. Varner, and H. Wollnik.Opportunistic mass measurements at the holifield radioactiveion beam facility. International Journal of Mass Spectrometry,251(2):119 – 124, 2006. ULTRA-ACCURATE MASS SPEC-TROMETRY AND RELATED TOPICS Dedicated to H.-J.Kluge on the occasion of his 65th birthday anniversary.

[HCJH77] J. C. Hardy, L. C. Carraz, B. Jonson, and P. G. Hansen. Theessential decay of pandemonium: A demonstration of errors incomplex beta-decay schemes. Physics Letters B, 71(2):307 – 310,1977.

[HHM+12] S. Hofmann, S. Heinz, R. Mann, J. Maurer, J. Khuyagbaatar,D. Ackermann, S. Antalic, W. Barth, M. Block, H. G. Burkhard,

143

BIBLIOGRAPHY

V. F. Comas, L. Dahl, K. Eberhardt, J. Gostic, R. A. Henderson,J. A. Heredia, F. P. Heßberger, J. M. Kenneally, B. Kindler,I. Kojouharov, J. V. Kratz, R. Lang, M. Leino, B. Lommel, K. J.Moody, G. Münzenberg, S. L. Nelson, K. Nishio, A. G. Popeko,J. Runke, S. Saro, D. A. Shaughnessy, M. A. Stoyer, P. Thörle-Pospiech, K. Tinschert, N. Trautmann, J. Uusitalo, P. A. Wilk,and A. V. Yeremin. The reaction 48Ca + 248Cm → 296116*studied at the gsi-ship. Eur. Phys. J. A, 48(5):62, 2012.

[HKLR+17] F. Heiße, F. Köhler-Langes, S. Rau, J. Hou, S. Junck, A. Kracke,A. Mooser, W. Quint, S. Ulmer, G. Werth, K. Blaum, andS. Sturm. High-Precision Measurement of the Proton’s AtomicMass. Phys. Rev. Lett., 119:033001, Jul 2017.

[HLMY+08] K. Hauschild, A. Lopez-Martens, A. V. Yeremin, O. Dorvaux,S. Antalic, A. V. Belozerov, Ch. Briançon, M. L. Chelnokov,V. I. Chepigin, D. Curien, B. Gall, A. Görgen, V. A. Gorshkov,M. Guttormsen, F. Hanappe, A. P. Kabachenko, F. Khalfal-lah, A. C. Larsen, O. N. Malyshev, A. Minkova, A. G. Popeko,M. Rousseau, N. Rowley, S. Saro, A. V. Shutov, S. Siem,L. Stuttgè, A. I. Svirikhin, N. U. H. Syed, Ch. Theisen, andM. Venhart. High-k, t1/2 = 1.4(1) ms, isomeric state in 255Lr.Phys. Rev. C, 78:021302, Aug 2008.

[HMV+82] S. Hofmann, G. Munzenberg, K. Valli, F. Heßberger, J. R. H.Schneider, P. Armbruster, B. Thuma, and Y. Eyal. Stopping ofalpha-recoil atoms in silicon. GSI-Report, page 241, 1982.

[Hof89] C. Hofmeyr. 53Cr(n, γ); transition energies and levels excited inthermal neutron capture. Nuclear Physics A, 500(1):111 – 126,1989.

[HSFM17] S. Hamzeloui, J. A. Smith, D. J. Fink, and E. G. Myers. Precisionmass ratio of 3He+ to HD+. Phys. Rev. A, 96:060501, Dec 2017.

[Hui54] J.R. Huizenga. Isotopic masses III. A > 201. Physica, 21(1):410– 424, 1954.

[IKKP80] M. A. Islam, T. J. Kennett, S. A. Kerr, and W. V. Prestwich.A self-consistent set of neutron separation energies. CanadianJournal of Physics, 58(2):168–173, 1980.

[INT07] INTERNATIONAL ATOMIC ENERGYAGENCY. Database ofPrompt Gamma Rays from Slow Neutron Capture for ElementalAnalysis. INTERNATIONAL ATOMIC ENERGY AGENCY,Vienna, 2007.

[ISW+13] Y. Ito, P. Schury, M. Wada, S. Naimi, T. Sonoda, H. Mita,F. Arai, A. Takamine, K. Okada, A. Ozawa, and H. Woll-nik. Single-reference high-precision mass measurement with a

144

BIBLIOGRAPHY

multireflection time-of-flight mass spectrograph. Phys. Rev. C,88:011306, Jul 2013.

[IUP14] IUPAC. Compendium of chemical terminology, 2nd ed. (the“gold book”). by a.d. mc, 2014.

[JAA+10] H. T. Johansson, Yu. Aksyutina, T. Aumann, K. Boretzky,M. J. G. Borge, A. Chatillon, L. V. Chulkov, D. Cortina-Gil,U. Datta Pramanik, H. Emling, C. Forssén, H.O.U. Fynbo,H. Geissel, G. Ickert, B. Jonson, R. Kulessa, C. Langer,M. Lantz, T. LeBleis, K. Mahata, M. Meister, G. Münzenberg,T. Nilsson, G. Nyman, R. Palit, S. Paschalis, W. Prokopowicz,R. Reifarth, A. Richter, K. Riisager, G. Schrieder, H. Simon,K. Sümmerer, O. Tengblad, H. Weick, and M. V. Zhukov. Theunbound isotopes 9,10He. Nuclear Physics A, 842(1):15 – 32,2010.

[JHJ84] A. S. Jensen, P. G. Hansen, and B. Jonson. New mass rela-tions and two- and four-nucleon correlations. Nuclear PhysicsA, 431(3):393 – 418, 1984.

[Jos62] B. D. Josephson. Possible new effects in superconductive tun-nelling. Physics Letters, 1(7):251 – 253, 1962.

[KAB+13] S. Kreim, D. Atanasov, D. Beck, K. Blaum, Ch. Böhm, Ch.Borgmann, M. Breitenfeldt, T.E. Cocolios, D. Fink, S. George,A. Herlert, A. Kellerbauer, U. Köster, M. Kowalska, D. Lunney,V. Manea, E. Minaya Ramirez, S. Naimi, D. Neidherr, T. Nicol,R. E. Rossel, M. Rosenbusch, L. Schweikhard, J. Stanja,F. Wienholtz, R. N. Wolf, and K. Zuber. Recent exploits ofthe ISOLTRAP mass spectrometer. Nuclear Instruments andMethods in Physics Research Section B: Beam Interactions withMaterials and Atoms, 317:492 – 500, 2013. XVIth InternationalConference on ElectroMagnetic Isotope Separators and Tech-niques Related to their Applications, December 2-7, 2012 atMatsue, Japan.

[Kar17] J. Karthein. Precision mass measurements using the Phase-Imaging Ion-Cyclotron-Resonance detection technique. MasterThesis, 2017.

[KBB+03] A. Kellerbauer, K. Blaum, G. Bollen, F. Herfurth, H.-J. Kluge,M. Kuckein, E. Sauvan, C. Scheidenberger, and L. Schweikhard.From direct to absolute mass measurements: A study of the ac-curacy of ISOLTRAP. Eur. Phys. J. D, 22(1):53–64, 2003.

[KBK+95] M. König, G. Bollen, H.-J. Kluge, T. Otto, and J. Szerypo.Quadrupole excitation of stored ion motion at the true cyclotronfrequency. International Journal of Mass Spectrometry and IonProcesses, 142(1):95 – 116, 1995.

145

BIBLIOGRAPHY

[KCL80] J. Kopecky, R.E. Chrien, and H.I. Liou. Resonance neutroncapture in 52cr. Nuclear Physics A, 334(1):35 – 44, 1980.

[KH88] K. S. Krane and D. Halliday. Introductory nuclear physics, vol-ume 465. Wiley New York, 1988.

[KLD+13] Z. Kohley, E. Lunderberg, P. A. DeYoung, A. Volya, T. Bau-mann, D. Bazin, G. Christian, N. L. Cooper, N. Frank, A. Gade,C. Hall, J. Hinnefeld, B. Luther, S. Mosby, W. A. Peters, J. K.Smith, J. Snyder, A. Spyrou, and M. Thoennessen. First obser-vation of the 13Li ground state. Phys. Rev. C, 87:011304, Jan2013.

[KMW58] T. P. Kohman, J. H. E. Mattauch, and A. H. Wapstra. NewReference Nuclide. Science, 127(3312):1431–1432, 1958.

[Kno12] G. F. Knoll. Radiation detection and measurement 4th Edition.John Wiley & Sons, 2012.

[KNT+16] Y. Kondo, T. Nakamura, R. Tanaka, R. Minakata, S. Ogoshi,N. A. Orr, N. L. Achouri, T. Aumann, H. Baba, F. Delaunay,P. Doornenbal, N. Fukuda, J. Gibelin, J. W. Hwang, N. In-abe, T. Isobe, D. Kameda, D. Kanno, S. Kim, N. Kobayashi,T. Kobayashi, T. Kubo, S. Leblond, J. Lee, F. M. Mar-qués, T. Motobayashi, D. Murai, T. Murakami, K. Muto,T. Nakashima, N. Nakatsuka, A. Navin, S. Nishi, H. Otsu,H. Sato, Y. Satou, Y. Shimizu, H. Suzuki, K. Takahashi,H. Takeda, S. Takeuchi, Y. Togano, A. G. Tuff, M. Vandebrouck,and K. Yoneda. Nucleus 26O: A Barely Unbound System beyondthe Drip Line. Phys. Rev. Lett., 116:102503, Mar 2016.

[Kön60] L. A. König. Mathematical Details of the Mass Computation.Proc. Int. Conf. Nuclidic Mases, pages 39–57, 1960.

[Kre07] M. Kretzschmar. The Ramsey method in high-precision massspectrometry with Penning traps: Theoretical foundations. In-ternational Journal of Mass Spectrometry, 264(2):122 – 145,2007.

[KRR16] A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team(2015). NIST Atomic Spectra Database (version 5.3), [On-line]. Available: http://physics.nist.gov/asd. National Instituteof Standards and Technology, Gaithersburg, MD., 2016.

[KSB+12] Z. Kohley, J. Snyder, T. Baumann, G. Christian, P. A. DeY-oung, J. E. Finck, R. A. Haring-Kaye, M. Jones, E. Lunderberg,B. Luther, S. Mosby, A. Simon, J. K. Smith, A. Spyrou, S. L.Stephenson, and M. Thoennessen. Unresolved Question of the10He Ground State Resonance. Phys. Rev. Lett., 109:232501, Dec2012.

146

BIBLIOGRAPHY

[KTUY05] H. Koura, T. Tachibana, M. Uno, and M. Yamada. Nuclidicmass formula on a spherical basis with an improved even-oddterm. Progress of Theoretical Physics, 113(2):305–325, 2005.

[KUTY00] H. Koura, M. Uno, T. Tachibana, and M. Yamada. Nuclear massformula with shell energies calculated by a new method. NuclearPhysics A, 674(1):47 – 76, 2000.

[LGR+05] Yu. A. Litvinov, H. Geissel, T. Radon, F. Attallah, G. Audi,K. Beckert, F. Bosch, M. Falch, B. Franzke, M. Hausmann,and M. Hellstr˙ Mass measurement of cooled neutron-deficientbismuth projectile fragments with time-resolved schottky massspectrometry at the frs-esr facility. Nuclear Physics A, 756(1):3– 38, 2005.

[LK12] Yu-Ran Luo and J. A. Kerr. Bond dissociation energies. CRCHandbook of Chemistry and Physics, 89, 2012.

[LM16a] P. J. Linstrom and W. G. Mallard. NIST Chemistry WebBook,NIST Standard Reference Database Number 69. National In-stitute of Standards and Technology, Gaithersburg MD, 20899,http://webbook.nist.gov, 2016.

[LM16b] A. Lopez-Martens. Private communication. April 2016.[LNST63] J. Lindhard, V. Nielsen, M. Scharff, and P.V. Thomsen. Integral

equations govering radiation effects. Mat. Fys. Medd. Vid. Selsk,33(10), 1963.

[LPT03] D. Lunney, J. M. Pearson, and C. Thibault. Recent trends inthe determination of nuclear masses. Rev. Mod. Phys., 75:1021–1082, Aug 2003.

[LS01] A. Lépine-Szily. Experimental overview of mass measurements.In David Lunney, Georges Audi, and H.-Jürgen Kluge, editors,Atomic Physics at Accelerators: Mass Spectrometry, pages 35–57, Dordrecht, 2001. Springer Netherlands.

[LT72] G. D. Loper and G. E. Thomas. Gamma-ray intensity standards:The reactions 14N(n, γ)15N, 35Cl(n, γ)36Cl and 53Cr(n, γ)54Cr.Nuclear Instruments and Methods, 105(3):453 – 460, 1972.

[MBC+13] A. G. Marshall, G. T. Blakney, T. Chen, N. K. Kaiser, A. M.McKenna, R. P. Rodgers, B. M. Ruddy, and F. Xian. Massresolution and mass accuracy: How much is enough? Mass Spec-trometry, 2, 2013.

[Mei15] Z. Meisel. Private communication. October 2015.[MES+12] M. Matos, A. Estradé, H. Schatz, D. Bazin, M. Famiano,

A. Gade, S. George, W. G. Lynch, Z. Meisel, M. Portillo,A. Rogers, D. Shapira, A. Stolz, M. Wallace, and J. Yurkon.Time-of-flight mass measurements of exotic nuclei. Nuclear

147

BIBLIOGRAPHY

Instruments and Methods in Physics Research Section A: Ac-celerators, Spectrometers, Detectors and Associated Equipment,696:171 – 179, 2012.

[MG13] Z. Meisel and S. George. Time-of-flight mass spectrometry ofvery exotic systems. International Journal of Mass Spectrome-try, 349-350:145 – 150, 2013. 100 years of Mass Spectrometry.

[MGA+15] Z. Meisel, S. George, S. Ahn, J. Browne, D. Bazin, B. A. Brown,J. F. Carpino, H. Chung, R. H. Cyburt, A. Estradé, M. Fami-ano, A. Gade, C. Langer, M. Matoš, W. Mittig, F. Montes, D. J.Morrissey, J. Pereira, H. Schatz, J. Schatz, M. Scott, D. Shapira,K. Smith, J. Stevens, W. Tan, O. Tarasov, S. Towers, K. Wim-mer, J. R. Winkelbauer, J. Yurkon, and R. G. T. Zegers. Massmeasurements demonstrate a strong N = 28 Shell Gap in argon.Phys. Rev. Lett., 114:022501, Jan 2015.

[MLH+16] D. A. Matters, A. G. Lerch, A. M. Hurst, L. Szentmiklósi, J. J.Carroll, B. Detwiler, Zs. Révay, J. W. McClory, S. R. McHale,R. B. Firestone, B. W. Sleaford, M. Krtička, and T. Belgya.Investigation of 186Re via radiative thermal-neutron capture on185Re. Phys. Rev. C, 93:054319, May 2016.

[MN88] P. Möller and J. R. Nix. Nuclear masses from a unifiedmacroscopic-microscopic model. Atomic Data and Nuclear DataTables, 39(2):213 – 223, 1988.

[MNMS95] P. Möller, J. R. Nix, W. D. Myers, and W. J. Swiatecki. Nu-clear ground-state masses and deformations. Atomic Data andNuclear Data Tables, 59(2):185 – 381, 1995.

[MNT16a] P. J. Mohr, D. B. Newell, and B. N. Taylor. Codata recom-mended values of the fundamental physical constants: 2014.Journal of Physical and Chemical Reference Data, 45(4):043102,2016.

[MNT16b] P. J. Mohr, D. B. Newell, and B. N. Taylor. Codata recom-mended values of the fundamental physical constants: 2014. Rev.Mod. Phys., 88:035009, Sep 2016.

[MSIS16] P. Möller, A.J. Sierk, T. Ichikawa, and H. Sagawa. Nuclearground-state masses and deformations: FRDM(2012). AtomicData and Nuclear Data Tables, 109-110(Supplement C):1 – 204,2016.

[MSS+18] T. D. Morris, J. Simonis, S. R. Stroberg, C. Stumpf, G. Ha-gen, J. D. Holt, G. R. Jansen, T. Papenbrock, R. Roth, andA. Schwenk. Structure of the lightest tin isotopes. Phys. Rev.Lett., 120:152503, Apr 2018.

[MTW65] J. H. E. Mattauch, W. Thiele, and A. H. Wapstra. 1964 Atomicmass table. Nuclear Physics, 67(1):1 – 31, 1965.

148

BIBLIOGRAPHY

[MWKW15] E. G. Myers, A. Wagner, H. Kracke, and B. A. Wesson. Atomicmasses of tritium and helium-3. Phys. Rev. Lett., 114:013003,Jan 2015.

[NBD+17] A. Negret, D. Balabanski, P. Dimitriou, Z. Elekes, T. J. Mertz-imekis, S. Pascu, and J. Timar. Nuclear structure and decaydata evaluation in europe. EPJ Web Conf., 146:02042, 2017.

[PBK+11] B. Pritychenko, E. Běták, M.A. Kellett, B. Singh, and J. Totans.The nuclear science references (NSR) database and web retrievalsystem. Nuclear Instruments and Methods in Physics ResearchSection A: Accelerators, Spectrometers, Detectors and Associ-ated Equipment, 640(1):213 – 218, 2011.

[Pea01] J. M. Pearson. The quest for a microscopic nuclear mass formula.Hyperfine Interactions, 132(1):59–74, Jan 2001.

[Pen01] Yu. E. Penionzhkevich. Mass Measurements in Nuclear Reac-tions, pages 265–273. Springer Netherlands, Dordrecht, 2001.

[Rat75] A. Ratkowski. Energy response of silicon surface-barrier particledetectors to slow heavy ions. Nuclear Instruments and Methods,130(2):533 – 538, 1975.

[rB96] Å. Björck. Numerical Methods for Least Squares Problems. So-ciety for Industrial and Applied Mathematics, 1996.

[RGG72] A. Rytz, B. Grennberg, and D. J. Gorman. New Alpha EnergyStandards, pages 1–9. Springer US, Boston, MA, 1972.

[RKA+04] D. Rodríguez, V. S. Kolhinen, G. Audi, J. Äystö, D. Beck,K. Blaum, G. Bollen, F. Herfurth, A. Jokinen, A. Kellerbauer,H. J. Kluge, M. Oinonen, H. Schatz, E. Sauvan, and S. Schwarz.Mass measurement on the rp-process waiting point 72Kr. Phys.Rev. Lett., 93:161104, Oct 2004.

[Ros02] L. Rosta. Cold neutron research facility at the budapest neutroncentre. Applied Physics A, 74(1):s52–s54, Dec 2002.

[RTP04] S. Rainville, J. K. Thompson, and D. E. Pritchard. An ion bal-ance for ultra-high-precision atomic mass measurements. Sci-ence, 303(5656):334–338, 2004.

[RW84] A. Rytz and R. A. P. Wiltshire. Absolute determination of theenergies of alpha particles emitted by 236Pu. Nuclear Instrumentsand Methods in Physics Research, 223(2-3):325 – 328, 1984.

[RWK86] A. Rytz, R. A. P. Wiltshire, and M. King. Absolute measure-ment of the energies of alpha-particles emitted by sources of252Cf and 227Ac. Nuclear Instruments and Methods in PhysicsResearch Section A: Accelerators, Spectrometers, Detectors andAssociated Equipment, 253(1):47 – 50, 1986.

149

BIBLIOGRAPHY

[SBB+91] G. Savard, St. Becker, G. Bollen, H.-J. Kluge, R. B. Moore, Th.Otto, L. Schweikhard, H. Stolzenberg, and U. Wiess. A newcooling technique for heavy ions in a Penning trap. PhysicsLetters A, 158(5):247 – 252, 1991.

[SBN+08] A. Solders, I. Bergström, Sz. Nagy, M. Suhonen, and R. Schuch.Determination of the proton mass from a measurement of thecyclotron frequencies of d+ and H2

+ in a Penning trap. Phys.Rev. A, 78:012514, Jul 2008.

[Sch13] H. Schatz. Nuclear masses in astrophysics. International Journalof Mass Spectrometry, 349-350:181 – 186, 2013. 100 years of MassSpectrometry.

[SKB+12] A. Spyrou, Z. Kohley, T. Baumann, D. Bazin, B. A. Brown,G. Christian, P. A. DeYoung, J. E. Finck, N. Frank, E. Lunder-berg, S. Mosby, W. A. Peters, A. Schiller, J. K. Smith, J. Snyder,M. J. Strongman, M. Thoennessen, and A. Volya. First Obser-vation of Ground State Dineutron Decay: 16Be. Phys. Rev. Lett.,108:102501, Mar 2012.

[SKL+08] B. Sun, R. Knöbel, Yu.A. Litvinov, H. Geissel, J. Meng, K. Beck-ert, F. Bosch, D. Boutin, C. Brandau, L. Chen, I.J. Cullen, C. Di-mopoulou, B. Fabian, M. Hausmann, C. Kozhuharov, S.A. Litvi-nov, M. Mazzocco, F. Montes, G. Münzenberg, A. Musumarra,S. Nakajima, C. Nociforo, F. Nolden, T. Ohtsubo, A. Ozawa,Z. Patyk, W.R. Plaß, C. Scheidenberger, M. Steck, T. Suzuki,P.M. Walker, H. Weick, N. Winckler, M. Winkler, and T. Ya-maguchi. Nuclear structure studies of short-lived neutron-richnuclei with the novel large-scale isochronous mass spectrometryat the FRS-ESR facility. Nuclear Physics A, 812(1):1 – 12, 2008.

[SLP18] A. Sobiczewski, Yu.A. Litvinov, and M. Palczewski. Detailed il-lustration of the accuracy of currently used nuclear-mass models.Atomic Data and Nuclear Data Tables, 119(Supplement C):1 –32, 2018.

[SLS+15] Y. P. Shen, W. P. Liu, J. Su, N. T. Zhang, L. Jing, Z. H. Li, Y. B.Wang, B. Guo, S. Q. Yan, Y. J. Li, S. Zeng, G. Lian, X. C. Du,L. Gan, X. X. Bai, J.S. Wang, Y. H. Zhang, X.H. Zhou, X.D.Tang, J. J. He, Y. Y. Yang, S.L. Jin, P. Ma, J. B. Ma, M. R.Huang, Z. Bai, Y. J. Zhou, W. H. Ma, J. Hu, S. W. Xu, S. B. Ma,S.Z. Chen, L. Y. Zhang, B. Ding, and Z. H. Li. Measurementof the 52Fe mass via the precise proton-decay energy of 53Com.Phys. Rev. C, 91:047304, Apr 2015.

[SMB+14] M. Del Santo, Z. Meisel, D. Bazin, A. Becerril, B. A. Brown,H. Crawford, R. Cyburt, S. George, G. F. Grinyer, G. Lorusso,P. F. Mantica, F. Montes, J. Pereira, H. Schatz, K. Smith, and

150

BIBLIOGRAPHY

M. Wiescher. β-delayed proton emission of 69Kr and the 68Se rp-process waiting point. Physics Letters B, 738:453 – 456, 2014.

[SXZ+16] P. Shuai, X. Xu, Y. H. Zhang, H. S. Xu, Yu. A. Litvinov,M. Wang, X. L. Tu, K. Blaum, X. H. Zhou, Y. J. Yuan, X. L.Yan, X. C. Chen, R. J. Chen, C. Y. Fu, Z. Ge, W. J. Huang,Y. M. Xing, and Q. Zeng. An improvement of isochronous massspectrometry: Velocity measurements using two time-of-flightdetectors. Nuclear Instruments and Methods in Physics ResearchSection B: Beam Interactions with Materials and Atoms, 376:311– 315, 2016. Proceedings of the XVIIth International Confer-ence on Electromagnetic Isotope Separators and Related Topics(EMIS2015), Grand Rapids, MI, U.S.A., 11-15 May 2015.

[Tay14] B. N. Taylor. Private communication. November 2014.[TBH+12] L. Trache, A. Banu, J.C. Hardy, V.E. Iacob, M. McCleskey,

B.T. Roeder, E. Simmons, A. Spiridon, R.E. Tribble, A. Saasta-moinen, A. Jokinen, J Äysto, T. Davinson, G. Lotay, P.J. Woods,and E. Pollacco. Decay spectroscopy for nuclear astrophysics: βand β-delayed proton decay. Journal of Physics: Conference Se-ries, 337(1):012058, 2012.

[Tul96] J. K. Tuli. Evaluated nuclear structure data file. Nuclear Instru-ments and Methods in Physics Research Section A: Accelerators,Spectrometers, Detectors and Associated Equipment, 369(2):506– 510, 1996.

[TXW+11] X. L. Tu, H. S. Xu, M. Wang, Y. H. Zhang, Yu. A. Litvinov,Y. Sun, H. Schatz, X. H. Zhou, Y. J. Yuan, J. W. Xia, G. Audi,K. Blaum, C. M. Du, P. Geng, Z. G. Hu, W. X. Huang, S. L.Jin, L. X. Liu, Y. Liu, X. Ma, R. S. Mao, B. Mei, P. Shuai, Z. Y.Sun, H. Suzuki, S. W. Tang, J. S. Wang, S. T. Wang, G. Q.Xiao, X. Xu, T. Yamaguchi, Y. Yamaguchi, X. L. Yan, J. C.Yang, R. P. Ye, Y. D. Zang, H. W. Zhao, T. C. Zhao, X. Y.Zhang, and W. L. Zhan. Direct mass measurements of short-lived a = 2z − 1 nuclides 63Ge, 65As, 67Se, and 71Kr and theirimpact on nucleosynthesis in the rp process. Phys. Rev. Lett.,106:112501, Mar 2011.

[VB72] D. Vautherin and D. M. Brink. Hartree-fock calculations withskyrme’s interaction. i. spherical nuclei. Phys. Rev. C, 5:626–647,Mar 1972.

[VCB68] G. B. Vingiani, G. Chilosi, and W. Bruynesteyn. The 49Caground-state analogue. Physics Letters B, 26(5):285 – 287, 1968.

[VDZS01] R. S. Van Dyck, S. L. Zafonte, and P. B. Schwinberg. Ultra-precise mass measurements using the uw-ptms. Hyperfine Inter-actions, 132(1):163–175, Jan 2001.

151

BIBLIOGRAPHY

[VWV+86] D. J. Vieira, J. M. Wouters, K. Vaziri, R. H. Kraus, H. Wollnik,G. W. Butler, F. K. Wohn, and A. H. Wapstra. Direct massmeasurements of neutron-rich light nuclei near N=20. Phys.Rev. Lett., 57:3253–3256, Dec 1986.

[WA85] A. H. Wapstra and G. Audi. The 1983 atomic mass evaluation:(i). atomic mass table. Nuclear Physics A, 432(1):1 – 54, 1985.

[WAA+17] A. Welker, N. A. S. Althubiti, D. Atanasov, K. Blaum,T. E. Cocolios, F. Herfurth, S. Kreim, D. Lunney, V. Manea,M. Mougeot, D. Neidherr, F. Nowacki, A. Poves, M. Rosenbusch,L. Schweikhard, F. Wienholtz, R. N. Wolf, and K. Zuber. Bind-ing energy of 79Cu: Probing the structure of the doubly magic78Ni from only one proton away. Phys. Rev. Lett., 119:192502,Nov 2017.

[WAH88] A. H. Wapstra, G. Audi, and R. Hoekstra. Atomic masses from(mainly) experimental data. Atomic Data and Nuclear DataTables, 39(2):281 – 287, 1988.

[WAK+17] M. Wang, G. Audi, F. G. Kondev, W. J. Huang, S. Naimi, andX. Xu. The ame2016 atomic mass evaluation (ii). tables, graphsand references. Chinese Physics C, 41(3):030003, 2017.

[Wap54a] A. H. Wapstra. Isotopic masses I. A < 34. Physica, 21(1):367 –384, 1954.

[Wap54b] A. H. Wapstra. Isotopic masses II. 33 < A < 202. Physica,21(1):385 – 409, 1954.

[Wap60] A. H. Wapstra. Treatment of input data for a least squares nu-clidic mass adjustment. Proc. Int. Conf. Nuclidic Mases, pages24–38, 1960.

[Wap65] A. H. Wapstra. Two-proton and two-neutron binding-energy sys-tematics and alpha-decay energies. Nuclear Data Sheets. SectionA, 1:1 – 19, 1965.

[WAW+12] M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev, M. Mac-Cormick, X. Xu, and B. Pfeiffer. The Ame2012 atomic massevaluation. Chinese Physics C, 36(12):1603, 2012.

[WBB+13a] F. Wienholtz, D. Beck, K. Blaum, Ch. Borgmann, M. Breiten-feldt, R. B. Cakirli, S. George, F. Herfurth, J. D. Holt, M. Kowal-ska, S. Kreim, D. Lunney, V. Manea, J. Menendez, D. Neid-herr, M. Rosenbusch, L. Schweikhard, A. Schwenk, J. Simonis,J. Stanja, and K. Zuber. Masses of exotic calcium isotopes pindown nuclear forces. Nature, 498:346–349, 2013.

[WBB+13b] R. N. Wolf, D. Beck, K. Blaum, Ch. Böhm, Ch. Borgmann,M. Breitenfeldt, N. Chamel, S. Goriely, F. Herfurth, M. Kowal-ska, S. Kreim, D. Lunney, V. Manea, E. Minaya Ramirez,

152

BIBLIOGRAPHY

S. Naimi, D. Neidherr, M. Rosenbusch, L. Schweikhard,J. Stanja, F. Wienholtz, and K. Zuber. Plumbing neutron starsto new depths with the binding energy of the exotic nuclide 82Zn.Phys. Rev. Lett., 110:041101, Jan 2013.

[Wei35] C. F. v. Weizsäcker. Zur theorie der kernmassen. Zeitschrift fürPhysik, 96(7):431–458, Jul 1935.

[WG71] A. H. Wapstra and N. B. Gove. Part i. atomic mass table. AtomicData and Nuclear Data Tables, 9(4):267 – 301, 1971.

[WGB68] D. H. White, D. J. Groves, and R. E. Birkett. Precision mea-surements of gamma rays from 60Co, 41Ar and 53Cr(n, γ)54Cr.Nuclear Instruments and Methods, 66(1):70 – 76, 1968.

[WKB77] A. H. Wapstra and K K. Bos. The 1977 atomic mass evaluation:in four parts. Atomic Data and Nuclear Data Tables, 20(1):1–125, 1977.

[WL11] N. Wang and M. Liu. Nuclear mass predictions with a radialbasis function approach. Phys. Rev. C, 84:051303, Nov 2011.

[WLWM14] N. Wang, M. Liu, X. Wu, and J. Meng. Surface diffusenesscorrection in global mass formula. Physics Letters B, 734(Sup-plement C):215 – 219, 2014.

[WMRS12] R. N. Wolf, G. Marx, M. Rosenbusch, and L. Schweikhard.Static-mirror ion capture and time focusing for electrostatic ion-beam traps and multi-reflection time-of-flight mass analyzers byuse of an in-trap potential lift. International Journal of MassSpectrometry, 313:8 – 14, 2012.

[WP90] H. Wollnik and M. Przewloka. Time-of-flight mass spectrometerswith multiply reflected ion trajectories. International Journal ofMass Spectrometry and Ion Processes, 96(3):267 – 274, 1990.

[WVW+85] J. M. Wouters, D. J. Vieira, H. Wollnik, H. A. Enge, S. Kowal-ski, and K. L. Brown. Optical design of the tofi (time-of-flightisochronous) spectrometer for mass measurements of exotic nu-clei. Nuclear Instruments and Methods in Physics Research Sec-tion A: Accelerators, Spectrometers, Detectors and AssociatedEquipment, 240(1):77 – 90, 1985.

[WWA+13] R. N. Wolf, F. Wienholtz, D. Atanasov, D. Beck, K. Blaum,Ch. Borgmann, F. Herfurth, M. Kowalska, S. Kreim, Yu. A.Litvinov, D. Lunney, V. Manea, D. Neidherr, M. Rosenbusch,L. Schweikhard, J. Stanja, and K. Zuber. ISOLTRAP’s multi-reflection time-of-flight mass separator/spectrometer. Interna-tional Journal of Mass Spectrometry, 349-350:123 – 133, 2013.100 years of Mass Spectrometry.

153

BIBLIOGRAPHY

[XZW+02] J. W. Xia, W. L. Zhan, B. W. Wei, Y. J. Yuan, M. T. Song,W. Z. Zhang, X. D. Yang, P. Yuan, D. Q. Gao, H. W. Zhao,X. T. Yang, G. Q. Xiao, K. T. Man, J. R. Dang, X. H. Cai, Y. F.Wang, J. Y. Tang, W. M. Qiao, Y. N. Rao, Y. He, L. Z. Mao,and Z. Z. Zhou. The heavy ion cooler-storage-ring project (hirfl-csr) at lanzhou. Nuclear Instruments and Methods in PhysicsResearch Section A: Accelerators, Spectrometers, Detectors andAssociated Equipment, 488(1):11 – 25, 2002.

[ZBM+82] J. D. Zumbro, C. P. Browne, J. F. Mateja, H. T. Fortune, andR. Middleton. 176Yb(t, p)178Yb reaction. Phys. Rev. C, 26:965–968, Sep 1982.

[ZJ15] S. L. Zafonte and R. S. Van Dyck Jr. Ultra-precise single-ionatomic mass measurements on deuterium and helium-3. Metrolo-gia, 52(2):280, 2015.

[ZXS+17] P. Zhang, X. Xu, P. Shuai, R. J. Chen, X. L. Yan, Y. H. Zhang,M. Wang, Yu. A. Litvinov, K. Blaum, H. S. Xu, T. Bao, X. C.Chen, H. Chen, C. Y. Fu, J. J. He, S. Kubono, Y. H. Lam, D. W.Liu, R. S. Mao, X. W. Ma, M. Z. Sun, X. L. Tu, Y. M. Xing, J. C.Yang, Y. J. Yuan, Q. Zeng, X. Zhou, X. H. Zhou, W. L. Zhan,S. Litvinov, G. Audi, T. Uesaka, Y. Yamaguchi, T. Yamaguchi,A. Ozawa, B. H. Sun, Y. Sun, and F. R. Xu. High-precisionQEC values of superallowed 0+ −→ 0+ β− emitters 46Cr,50 Feand 54Ni. Physics Letters B, 767:20–24, 2017.

154

Synthèse

Les masses atomiques ont été largement utilisées en physique nucléaire,astrophysique et dans des applications avancées telles que l’énergie nucléaire etla gestion des déchets. La masse atomique est une empreinte digitale uniqued’un noyau. En mesurant la masse atomique d’un noyau, nous pouvons endériver l’énergie de liaison, qui reflète toutes les interactions (fortes, faibles etélectromagnétiques) en action dans le noyau. L’évaluation des masses atom-iques (AME), commencée dans les années 1950, vise à fournir l’information laplus fiable et la plus complète sur les masses atomiques. Jusqu’à présent, dixtables de masse ont été publiées sur la base de la méthode de Wapstra.

Les connaissances actuelles des masses atomiques peuvent être obtenuespar quatre voies: a) les énergies de désintégration bêta, b) les énergies dedésintégration dues aux émissions de particules légères, e.g., α et proton, c)les énergies libérées dans les réactions nucléaires, et d) les données de spec-trométrie de masse. Les trois premières méthodes sont indirectes car ellesmesurent l’énergie libérée, et la dernière est directe, car elle mesure la masseinertielle d’un atome dans un champ électromagnétique. Toutes ces mesures demasse sont relatives, ce qui signifie que chaque donnée expérimentale établitfondamentalement une relation entre deux masses, ou parfois plusieurs. Enmême temps, puisque le nombre de mesures est beaucoup plus grand que celuides masses, extraire les masses de ce système surdéterminé n’est pas sim-ple. Sur la base de ces faits, l’évaluation des masses atomiques est soumise àun traitement spécial des données. Pour résoudre le problème de la surdéter-mination, nous pouvons recourir à la méthode des moindres carrés. Puisquetoutes les données d’entrée sont linéaires en masse, nous pouvons considérerles masses comme des paramètres dans la méthode des moindres carrés. Untel système intriqué peut ainsi être résolu, et les masses peuvent être dérivéessans approximation. L’utilisation de la méthode des moindres carrés sur lesystème surdéterminé est une procédure idéale dans la mesure où elle fournitnon seulement des valeurs de masse non biaisées et fiables dérivées de donnéesexpérimentales, mais permet également de vérifier les consistances de toutesles données d’entrée. L’un des rôles de l’AME est de révéler des erreurs systé-matiques non découvertes, en comparant les données d’entrée avec les valeursajustées dans un adjustment global. Une telle tâche ne peut être effectuée quedans le cadre de l’AME. Après avoir obtenu la meilleure valeur pour chaque

155

masse, nous pouvons calculer toute combinaison de différences de masse, tellesque les énergies de désintégration et de séparation, sur la base de la matricede covariance. Pour trouver l’information qu’une équation apporte à chaquemasse, on peut construire une matrice de flux d’information, découverte par G.Audi en 1986. Cette matrice permet de déterminer la contribution de chaquedonnée d’entrée sur chaque masse.

Les développements pour la dernière table de masse AME2016 sont dis-cutés. Le premier est lié au traitement soigneux des données les plus précises.Les données de masse les plus précises proviennent de la spectrométrie demasse à piège de Penning. De nos jours, comme la précision des pièges Penningpeut atteindre 10−10 ou même mieux, les énergies de liaison moléculaire et élec-tronique doivent être prises en compte. Une méthode pour calculer l’énergie deliaison moléculaire à partir de la chaleur de formation standard est décrite etdeux exemples détaillés sont donnés. Nous trouvons qu’en utilisant la chaleurde formation standard mise à jour, les valeurs recalculées pour certaines desdonnées de spectrométrie de masse peuvent changer de manière significative.En AME2016, toutes les données précises ont été recalculées.

Le second est liée aux corrections des énergies de désintégration. Pour lesénergies de décroissance α et proton mesurées par des méthodes d’implantation,l’énergie de recul du partenaire de la décroissance doit être prise en comptecorrectement. Nous présentons une procédure pour corriger les énergies de dés-intégration publiées dans le cas où le recul du noyau n’est pas prise en comptedans les expériences d’implantation. Un programme a été développé basé surla théorie intégrale de Lindhard, qui prédit avec précision le dépôt d’énergiedes noyaux lourds dans la matière. Trois exemples sont donnés pour illustrerla procédure de correction.

Le troisième développement concerne la prise en compte de l’effet rela-tiviste et de l’effet atomique dans la conversion des énergies des particulesalpha en énergie de décroissance. Les énergies de décroissance α les plus pré-cises proviennent de la spectrographie magnétique. Ces étalons d’énergie αservent non seulement de points d’étalonnage pour tous les spectres α avec unpouvoir de résolution élevé, mais fournissent également des valeurs d’entréeprécises à l’AME. Afin d’obtenir des énergies de décroissance correctes à par-tir de la spectrographie magnétique, où seules les particules α sont détectées,une formule relativiste prenant en compte également les énergies de liaison desélectrons de l’hélium est dérivée.

Les modèles de masse sont le dernier recours pour accéder aux nucléidesles plus exotiques qui ne peuvent pas être produit actuellement, ni dans unproche avenir. Ces modèles de masse, indépendamment de leurs caractèresintrinsèques, peuvent reproduire les masses connues expérimentalement maisprédire un comportement différent lorsqu’elles sont extrapolées vers des régionsinconnues. Sur la base de ce fait, la précision et la puissance prédictive de huitmasses des modèles de divers types, à savoir la méthode de Thomas-Fermiétendue plus l’intégrale de Strutinsky (ETFSI-2), le modèle de gouttelettes à

156

portée finie (FRDM95) et sa version mise à jour avec traitement amélioré dela déformation (FRDM12), un modèle récent Weizsäcker-Skyrme plus RadialBasic Function (WS4+RBF), deux modèles récents de masse Hartree-Fock-Bogoliubov HFB26 et HFB27, le modèle Duflo et Zuker (DUZU), le modèleKTUY05, sont étudiés. Nous trouvons que la déviation quadratique moyenne(δrms) de tous les modèles de masse considérés est bien inférieure à 0.8 MeVpar rapport aux trois tables de masse AME2003, AME2012 et AME2016 pourtous les nucléides (N,Z ≥ 8). Le modèle de masse WS4+RBF est le modèlede masse le plus précis qui donne δrms autour de 0.2 MeV. Le modèle de massemicroscopique HFB27 présente également une bonne précision avec δrms ≈0.5 MeV. Le modèle de masse DUZU est encore un modèle de masse robustequi donne δrms ≈ 0.4 MeV. Les écarts quadratiques très similaires pour chaquemodèle de masse par rapport aux différentes tables de masse sont dus au faitque, dans AME2016, seules 17 masses ont changé de plus de 500 keV parrapport à AME2003.

L’extrapolation de masse d’AME est une autre façon d’obtenir des massesinconnues. La régularité est considérée comme une propriété fondamentale dela surface de masse et peut aider à dériver des masses inconnues à partir desmasses mesurées. Les représentations directes des masses dans un espace tridi-mensionnel ne sont pas pratiques, car la surface de masse a une très grandeextension le long de l’axe de la masse (nombre de masse allant de 1 u à environ300 u). Dans le travail de [WAH88], l’extrapolation a été faite en examinant lasurface de la masse en quatre projections. Wapstra et al. [WAH88] ont analyséles différences entre masses expérimentales et une expression obtenue par unexamen approprié des effets d’appariement [JHJ84] plus une fonction lisse.Un tel traitement a permis d’examiner la surface de masse à peu près à lamême échelle (plusieurs MeV) et remédier à certaines oscillations causées parl’appariement. Cependant, la procédure était plutôt cpmplexe (voir Fig. 1 dans[WAH88]). Une alternative pour contourner ces difficultés est d’examiner lesdérivées de la surface de la masse. Par dérivée, nous entendons une différencespécifique entre les masses de deux nucléides voisins. Les dérivés conserventle comportement lisse qui s’étend des masses connues aux inconnues d’unepart, et amplifier la structure locale d’un autre côté. Le travail de pionnierde [Wap65] vise à estimer des masses inconnues sur la base des études decinq dérivés: énergies de séparation à deux neutrons et à deux protons, éner-gies de désintégration α, désintégration bêta et les énergies de désintégrationdouble-bêta. L’intention première de l’extrapolation de mass était d’exploterles energies de réaction et de décroissance entre noyaux n’ayant aucun lien avecles noyaux de masses connues. Une masse suspecte qui perturbe les graphiquesdes dérivées peut également être repérée facilement.

Un modèle de masse peut également être utile dans l’extrapolation demasse. Nous soustrayons les masses prédictes par un modèle de masse desmasses expérimentales et étudions la tendance des écarts. Treize modèlesde masse ont été étudiés [BA93] et le modèle de DUZU à base sphérique

157

(DZ10sph) [DZ96] a été choisi comme modèle préféré, car sa surface de massepeut être affichée beaucoup plus en douceur avec ce modèle. En supposant quela tendance des écarts est régulière et continue, nous pourrions étendre cettetendance des masses connues aux masses inconnues.

La combinaison de produits dérivés et l’utilisation de la différence en-tre les masses et les masses d’un modèle permet une extrapolation pratique.Comme chaque point dans les graphiques des dérivées implique deux masseset que l’on doit trouver lequel est le responsable du point de déraillementdans tout graphiques des dérivées. Si l’on travaille sur le graphique de ladifférence de masse, on peut manipuler chaque masse plus facilement. Ledéveloppement d’un outil graphique interactif dans les années 1990, qui estencore utilisé aujourd’hui, nous permet de vérifier quatre types différents desurface de masse en même temps. Tout changement d’une masse unique dansun graphique de la surface de masse spécifique mettra à jour l’informationdans les autres graphiques. Et un tel changement devrait être cohérent dansles quatre graphiques. L’extrapolation de masse fournit les meilleures esti-mations pour les masses inconnues qui ne sont pas trop loin (deux ou troisunités de masse) des derniers nucléides connus. Cette méthode est basée surla douceur de la surface de masse et une telle caractéristique lisse devrait êtrepréférée lorsque nous extrapolons les masses vers la région inconnue. L’écartmoyen-carré pour toutes les masses estimées dans l’AME2012 qui sont connuesdans l’AME2016 est de 0.396 MeV (55 cas au total), ce qui est plus petit quetous les modèles de masse discutés ici.

Une autre partie de cette thèse est liée aux mesures de masse effec-tuées en utilisant le spectromètre de masse à piège de Penning ISOLTRAPà ISOLDE/CERN. Le confinement spatial complet exige un minimum de po-tentiel dans les trois dimensions. Une force de confinement souhaitable est cellequi dépend linéairement de la distance entre les ions stockés et le centre dupiège. Cela entraîne un potentiel harmonique pour les ions confinés. Cepen-dant, ni un champ magnétique pur ni un champ électrostatique ne peuventconfiner un ion en 3-dimensions. Une superposition d’un champ magnétiquehomogène fort, assurant un confinement radial, et un champ quadripolaireélectrostatique faible, assurant un confinement axial, est utilisé pour attein-dre le confinement tridimensionnel dans le piège de Penning. Les ions danscette combinations de champs effectuent trois mouvements propres indépen-dants [BG86]: l’oscillation harmonique le long de l’axe du piège à la fréquenced’oscillation axiale ωz, le mouvement cyclotron modifié à la fréquence ω+ etle magnétron à la fréquence ω−. Les deux derniers mouvements sont des mou-vements radiaux et perpendiculaires à l’axe du piège. En raison des imperfec-tions du vrai piège de Penning, tels que les trous d’ouverture pour l’injection etl’éjection d’ions, ou les électrodes ne s’étendant pas à l’infini, ce qui induiraitun champ électrique multipolaire d’ordre supérieur, ce qui rend ωc = ω+ +ω−invalide. Basé sur ce fait, le refroidissement est important pour effectuer desmesures de haute précision, puisque les ions sondent moins les imperfections

158

des champs électriques et magnétiques. Pour les nucléides radioactifs, le re-froidissement du gaz tampon est généralement utilisé. Puisque les gaz noblesont un potentiel d’excitation élevé, ils constituent un choix idéal pour refroidirles ions. Par collisions avec des atomes de gaz rares, tels que l’hélium, lesions chauds perdent de l’énergie et leur mouvement peut être amorti. Cepen-dant, la situation dans les pièges Penning devient plus complexe. Pendantle refroidissement, les amplitudes des mouvements cyclotron axial et modifiédiminuent alors que l’amplitude du mouvement magnétron augmente. Finale-ment, les ions heurtent l’électrode annulaire et sont perdus. Pour contrer ladiffusion radiale vers l’extérieur provoquée par le gaz tampon, on peut utiliserla technique de refroidissement par bandes latérales [SBB+ 91]. En appliquantune tension RF à l’électrode annulaire segmentée à la fréquence cyclotronωc = ω+ + ω−, les mouvements radiaux se couplent, c’est-à-dire que la con-version du mouvement cyclotron modifié en mouvement magnétron aura lieupériodiquement. Puisque le mouvement cyclotron modifié est amorti plus rapi-dement que le mouvement magnétron, les ions peuvent être recentrés après uncertain temps de refroidissement. Le refroidissement par bandes latérales estsélectif en masse et peut être utilisé pour recentrer les ions d’intérêt et éliminerles contaminations isobariques.

Une excitation quadripolaire à la somme des fréquences propres individu-elles peut être utilisée pour coupler des mouvements propres et déterminerdes fréquences. L’approche la plus directe pour la spectrométrie de masse estla mesure de la fréquence somme ωc = ω+ + ω−. Le couplage des deux mou-vements radiaux peut être réalisé par un champ rf azimutal quadrupolaire ála fréquence ωrf appliquée avec des déphasages de 180◦ sur des ensembles desegments d’électrodes annulaires perpendiculaires l’un à l’autre. En résonance,ωrf = ωc, on obtient une conversion périodique complète entre les deux mou-vements. Dans les expériences, nous scannons la fréquence rf en rechargeant eten vidant le piège de Penning et en détectant le temps de vol d’un ion éjectédepuis le centre du piège. Le minimum dans le spectre de temps de vol nousdonne la fréquence cyclotron de l’ion.

Les masses de dix-huit nucléides provenant de plusieurs campagnes ex-périmentales entre 2011 et 2016 sont analysées. La masse de 168Lu dans sonétat isomérique a été mesurée et sa valeur est conforme à la valeur recom-mandée dans AME2012 mais neuf fois plus précise. La valeur de masse de178Yb obtenue ici diffère de 31 (13) keV du résultat d’une mesure par réactionantérieuse, mais confirme la liaison supplémentaire de 178Yb de 440 keV. Unaplatissement soudain dans S2n de 178Yb indiquerait l’existence d’une transi-tion de phase dans la région. Cependant, plus de mesures sont nécessaires pourclarifier cette divergence. Les masses de certains nucléides de terres rares telsque 140Nd et 160Yb sont mesurées avec une plus grande précision par rapportà AME2016. Les résultats d’autres nucléides aident également à améliorer laprécision des masses existantes.

Dans une autre étude, l’erreur systématique des mesures avec un spec-

159

tromètre de masse à multi-réflexion á temps de vol d’ISOLTRAP (MR-TOFMS) à ISOLTRAP a été étudiée en utilisant des sources d’ions hors ligne et avefaisceaux de protons. Dans l’étude hors ligne, différents paramètres sur MR-TOF MS ont été sondés. Deux des conclusions les plus importantes sont abor-dées. Premièrement, les mesures effectuées au nombre de réflexion N = 100n’est pas fiable. Afin de minimiser l’écart, les nombres de réflexion supérieursà 100 doivent être utilisés à la place. Deuxièmement, si nous utilisons dessources d’ions hors ligne pour l’étalonnage de masse, une incertitude de 170keV devrait être ajoutée au résultat final. Pour l’étude en ligne, 47 mesuressont sélectionnées à partir des mesures de contrôle de rendement. Le chi-carréréduit χn est déduit à 1.02, ce qui signifie que l’erreur systématique est beau-coup plus petite que les incertitudes statistiques. L’effet du second paramètreβ dans la formule de détermination de la masse est également étudié. Le ré-sultat montre que l’utilisation d’un seul ion de référence n’affectera pas pourles doublets de masse la détermination finale des valeurs de masse et donc laconclusion ci-dessus en ce qui concern l’erreur systématique.

160

Titre : Mesures Directes de Masses et Évaluation Globale des Masses Atomiques

Mots clefs : Évaluation des masses atomiques (Ame), Masse atomique, Méthode des moin-dres carrés, Spectrométrie de masse

Résumé :

L’évaluation des masses atomiques (Ame), commencée dans les années 1960, est la source la plus fiabled’informations complètes sur les masses atomiques. Elle fournit les meilleures valeurs pour les massesatomiques et les incertitudes associées en évaluant les données expérimentales de désintégration, deréactions et de la spectrométrie de masse.Dans cette thèse, la philosophie et les caractéristiques les plus importantes de l’Ame seront discutéesen détail. Les développements les plus récents de l’évaluation, AME2016, tels que l’énergie de liaisonmoléculaire, la correction d’énergie des mesures par implantation, et la formule relativiste pour le pro-cessus de décroissance alpha, seront présentés.Une autre partie de cette thèse concerne l’analyse des données du spectromètre à piège de PenningISOLTRAP au ISOLDE/CERN. Les nouveaux résultats sont inclus dans l’ajustement global et leursinfluences sur les masses existantes sont discutées.La dernière partie de cette thèse porte sur les études des erreurs systématiques du spectromètre de masseà multi-réflexion à temps de vol d’ISOLTRAP, utilisant une source d’ions hors ligne et le faisceau deprotons en ligne. A partir de l’analyse des mesures sélectionnées, j’ai trouvé que l’erreur systématiqueest beaucoup plus faible que les incertitudes statistiques obtenues jusqu’à présent.

Title : Direct Mass Measurements and Global Evaluation of Atomic Masses

Keywords : Atomic Mass Evaluation (Ame), Atomic Mass, Least-squares Method, MassSpectrometry

Abstract :

The Atomic Mass Evaluation (Ame), started in the 1960s, is the most reliable source for comprehensiveinformation related to atomic masses. It provides the best values for the atomic masses and theirassociated uncertainties by evaluating experimental data from decay, reactions, and mass spectrometry.In this thesis, the philosophy and the most important features of the Ame will be discussed in detail.The most recent developments of the latest mass table (AME2016), such as molecular binding energy,energy correction of the implantation measurements, and the relativistic formula for the alpha-decayprocess, will be presented.Another part of this thesis concerns the data analysis from the Penning-trap spectrometer ISOLTRAPat ISOLDE/CERN. The new results are included in the global adjustment and their influences on theexisting masses are discussed.The last part of this thesis is related to the systematic error studies of the ISOLTRAP multi-reflectiontime-of-flight mass spectrometer, using an off-line ion source and the on-line proton beam. From theanalysis of the selected measurements, I found that the systematic error is much smaller than thestatistical uncertainties obtained up to now.

Université Paris-SaclayEspace Technologique / Immeuble DiscoveryRoute de l’Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

161

direct mass measurements and global evaluation of atomic...

Documents